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Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker–Planck Equations

  • Tao Sun
  • Tian-jun WangEmail author
Original Paper
  • 80 Downloads

Abstract

Results on the composite generalized Laguerre–Legendre interpolation in unbounded domains are established. As an application, a composite Laguerre–Legendre pseudospectral scheme is presented for nonlinear Fokker–Planck equations on the whole line. The convergence and the stability of the proposed scheme are proved. Numerical results show the efficiency of the scheme and conform well to theoretical analysis.

Keywords

Composite generalized Laguerre–Legendre pseudospectral approximation Nonlinear Fokker–Planck equations defined on unbounded domains Multi-domain decomposition pseudospectral method 

Mathematics Subject Classification

65M70 41A30 35K55 

1 Introduction

Recently, Guo [15, 16], Shen and Wang [35] reviewed some results on the spectral methods for problems defined on unbounded domains. In general, there are two types of techniques (cf. Guo [16], Shen and Wang [35]):

(i) approximate the solution of problems defined on unbounded domains by classical orthogonal systems, such as Laguerre or Hermite polynomials/functions (see e.g., [1, 3, 5, 11, 13, 18, 22, 32, 33]). Also, some authors considered nonlinear problems defined on unbounded domains using the aforementioned methods (see e.g., [14, 41, 42, 43]);

(ii) approximate the solution of problems defined on bounded/unbounded domains by non-classical orthogonal systems (see e.g., [5]), or by mapped orthogonal systems, e.g., the image of classical Jacobi polynomials through a suitable mapping (see e.g., [19, 20, 21, 39]).

For the partial differential equations of non-standard types, we prefer to use composite orthogonal approximations and interpolation methods of the Laguerre (Hermite) polynomials/functions and Jacobi polynomials with the domain decomposition; see [8, 16, 26, 38] and the references therein.

Since the classical Laguerre (with \(\alpha =0\)) and Hermite functions are orthogonal in the non-weighted Sobolev spaces, we prefer to take them as basis for self-adjoint problems with constant or polynomial coefficients, which may lead to symmetric systems with computationally efficient sparse systems. Moreover, we can greatly improve the numerical solution accuracy by choosing adjustable parameters involved in the Laguerre or Hermite functions. In other words, the choice of the scaling parameters is meticulous for Laguerre and Hermite methods; see [35]. For cases of certain nonlinear problems defined on unbounded domains; see [6, 10, 11, 17, 18, 23, 31, 34, 36, 40, 43, 44]. Wang [37] considered spectral methods for the nonlinear Fokker–Planck equation defined on the whole line using the scaled generalized Laguerre functions coupled with the domain decomposition. The scaling parameter \(\beta\) offers great flexibility to match the asymptotic behaviors of exact solutions for differential equations of degenerate type at \(|v|\rightarrow \infty\). Whereas the solutions may vary rapidly near the fixed boundaries or in the corresponding bounded subdomains, the above framework is no longer the most appropriate; see [17, 44].

Let \(v\in \mathbb {R}:=(-\infty ,\infty )\) be the velocity of particles. Denote by W(vt) the probability density. The parameter \(k=-1\) or 1. Moreover, \(W_0(v)\) is the initial state. For simplicity, let \(\partial _z W=\dfrac{\partial W}{ \partial z},\) etc. In this paper, we consider the Fokker–Planck equations modeling the relaxation of fermion and boson gases (cf. [37]):
$$\begin{aligned} \left\{ \begin{array}{ll} \partial _tW(v,t)=\partial _v(vW(v,t)(1+kW(v,t)))+ \partial _v^2 W(v,t), \qquad &{}v\in \mathbb {R},\,\,0<t\le T,\\ W(v,t)\rightarrow 0, &{} |v|\rightarrow \infty ,\,\,0<t\le T,\\ W(v,0)=W_0(v),&{}v\in \mathbb {R}. \end{array}\right. \end{aligned}$$
(1)
These Fokker–Planck equations have been proposed in [4, 9, 29] and the references therein, as kinetic models for the relaxation to equilibrium of bosons (\(k=1\)) and fermions (\(k=-1\)). These models have been introduced as a simplification with respect to Boltzmann-based models; see [7, 30]. There is an integrable stationary solution of problem (1) (cf. [2]), \(F_\infty (v)= \big (\beta \text{e}^{ \frac{v^2}{2}}-k\big )^{-1}\), which shows that the solution decays very fast as \(v\rightarrow \infty\). For the same level of solution accuracy, the Laguerre spectral scheme requires more regularities of W(vt) than the Legendre spectral scheme in bounded domains; see Remark 4.1 of this work, or Remark 4.1 of [17]. Nevertheless, when the solutions of the underlying problems are oscillatory in a certain bounded subdomain or the solutions may vary rapidly near the fixed boundaries, Legendre polynomials are better than Laguerre functions; see [17, 44]. On the other hand, the solutions of the Fokker–Planck equation in statistical physics and the Black–Scholes equation in financial mathematics usually present the Gaussian distribution. In such cases, how to simulate the solution in the bounded subdomains near the peak value is essential. Thus, we prefer the multi-domain decomposition method for solving problem (1), such as the composite Laguerre–Legendre spectral or pseudospectral method on the whole line.

This paper is devoted to developing the composite generalized Laguerre–Legendre interpolation approximation and solving problem (1) using the multi-domain decomposition pseudospectral method. To do this, let \(\Lambda _1=(1,\infty ), \Lambda _2=(-\infty ,-1)\), \(I=(-1,1)\) and \(\mathbb {R}={\Lambda }_1 \cup {\overline{I}} \cup {\Lambda }_2\). Then, we approximate W(vt) on I by the Legendre pseudospectral method, and approximate W(vt) on \(\Lambda _1\) and \(\Lambda _2\) by the Laguerre pseudospectral method. Comparing the pseudospectral method with the spectral method, numerical solutions can be matched more conveniently on the common boundary of adjacent subdomains and nonlinear terms can be handed much more easily. Also, we can decrease the mode N of the Laguerre pseudospectral method. Since W(vt) decays very fast as \(v\rightarrow \infty\), Laguerre functions work better as the basis in actual computation on \(\Lambda _j\); see [28], which are \(L^2(\Lambda _j)\)-orthogonal systems. The use of scaled generalized Laguerre functions leads to much simplified analysis and more precise error estimates. Thus, we construct a Laguerre–Legendre pseudospectral scheme for problem (1) and prove its spectral accuracy in space. We also design an efficient algorithm for actual computation and numerical results show its efficiency.

This paper is organized as follows. In Sect. 2, we establish some results on the composite generalized Laguerre–Legendre interpolation approximation, which play important roles in the numerical analysis of pseudospectral methods for various differential equations on unbounded domains. We describe the algorithm, present numerical results and prove its convergence and stability in Sect. 3. The final section is for some concluding remarks.

2 Composite Laguerre–Legendre Interpolations

In this section, we establish some basic results on the composite Laguerre–Legendre interpolation approximations.

2.1 Generalized Laguerre–Gauss–Radau Interpolation

Denote \(\mathbb {R}^+:=(0,\infty )\), and let \(\chi (v)\) be a certain weight function. For any integer \(r\ge 0\),
$$\begin{aligned} H_{\chi }^{r}(\mathbb {R}^+)=\{u\,|\,u\,\mathrm{is\,measurable\,on} \,{\mathbb {R}^+\,\mathrm{and} \,\,||u||_{r,\chi ,\mathbb {R}^+}}<\infty \}, \end{aligned}$$
equipped with the following inner product, semi-norm and norm:
$$\begin{aligned} (u,w)_{r,\chi ,\mathbb {R}^+}= & {} \sum _{0\le k \le r}\int _0^\infty \partial _v^k u(v)\partial _v^k w(v)\chi (v)\text {d}v,\\ |u|_{r,\chi ,\mathbb {R}^+}= & {}\left (\int _0^\infty (\partial _v^r u(v))^2\chi (v)\text {d}v\right)^{\frac12},\quad \quad ||u||_{r,\chi ,\mathbb {R}^+}=(u,u)_{r,\chi ,\mathbb {R}^+}^{\frac{1}{2} }. \end{aligned}$$
In particular, \(H_{\chi }^0(\mathbb {R}^+)=L_{\chi }^2({\mathbb {R}^+})\), with the inner product \((u,w)_{\chi ,\mathbb {R}^+}\) and the norm \(||u||_{\chi ,\mathbb {R}^+}\). We omit the subscript \(\chi\) in the notations when \(\chi (v)\equiv 1.\) Throughout the paper, we follow the same line as the above definition on semi-norm and norm to define other weighted semi-norms and norms.
Let \(\omega _{\alpha ,\beta }(v)=v^{\alpha }\text{e}^{-\beta v},\)\(\alpha>-1, \beta >0\). The generalized Laguerre polynomial of degree l is defined by
$$\begin{aligned} \mathcal{L}_{l}^{(\alpha ,\beta )}(v)={\frac{1}{l!}}v^{-\alpha } \text{e}^{\beta v} \partial _v^{l}(v^{l+\alpha }\text{e}^{-\beta v}). \end{aligned}$$
The set of \(\mathcal{L}_{l}^{(\alpha ,\beta )}(v)\) is a complete \(L^2_{\omega _{\alpha ,\beta }}({\mathbb {R}^+})\)-orthogonal system.
For the problem whose solution decays very fast as \(v\rightarrow \infty\), we prefer the orthogonal system of generalized Laguerre functions, defined by
$$\begin{aligned} \tilde{\mathcal{L}}_{l}^{(\alpha ,\beta )}(v)=\text{e}^{-{\frac{1}{2}}\beta v}\mathcal{L}_{l}^{(\alpha ,\beta )}(v), \quad l=0,1,2,\ldots. \end{aligned}$$
(2)
The set of \(\tilde{\mathcal{L}}_{l}^{(\alpha ,\beta )}(v)\) is a complete \(L^2_{v^{\alpha }}(\mathbb {R}^+)\)-orthogonal system.
Let \(\Lambda _1\) as the same as before and \(\omega ^1_{\alpha ,\beta }(v)=\omega _{\alpha ,\beta }(v-1)\). We denote by \(\mathcal{P}_{N}(\Lambda _1)\) the set of all polynomials of degree at most N. Let \(\xi ^{(\alpha ,\beta )}_{R,N,\Lambda _1, j}, \, 0 \le j \le N,\) be the zeros of polynomial \((v-1)\partial _v\mathcal{L}_{N+1}^{(\alpha ,\beta )}(v-1)\), which are arranged in ascending order. Denote by \(\omega ^{(\alpha ,\beta )}_{R, N, \Lambda _1, j},\, 0 \le j \le N,\) the corresponding Christoffel numbers such that
$$\begin{aligned} \int _{\Lambda _1}\phi (v)\omega ^1_{\alpha ,\beta }(v)\text {d}v=\sum _{j=0}^{N}\phi (\xi ^{(\alpha ,\beta )}_{R,N,\Lambda _1, j}) \omega ^{(\alpha ,\beta )}_{R,N, \Lambda _1, j},\quad \forall \phi \in \mathcal{P}_{2N}(\Lambda _1). \end{aligned}$$
(3)
We now consider the new generalized Laguerre–Gauss–Radau interpolation corresponding to the weight function \((v-1)^\alpha\). Let \({\tilde{\xi }}^{(\alpha ,\beta )}_{R,N,\Lambda _1, j}=\xi ^{(\alpha ,\beta )}_{R,N,\Lambda _1,j}\) and \({\tilde{\omega }}^{(\alpha ,\beta )}_{R,N,\Lambda _1, j}=\text{e}^{\beta \xi ^{(\alpha ,\beta )}_{R,N,\Lambda _1, j}}\omega ^{(\alpha ,\beta )}_{R,N,\Lambda _1, j},\, 0 \le j \le N.\)
We introduce the following discrete inner product and norm (cf. [38]):
$$\begin{aligned} (u,w)_{(v-1)^{\alpha }, N,\Lambda _1}=\sum \limits _{j=0}^{N}u({\tilde{\xi }}^{(\alpha ,\beta )}_{R,N,\Lambda _1,j}) w({\tilde{\xi }}^{(\alpha ,\beta )}_{R,N,\Lambda _1,j}){\tilde{\omega }}^{(\alpha ,\beta )}_{R,N,\Lambda _1,j}, \quad ||u||_{(v-1)^{\alpha },N, \Lambda _1}=(u,u)_{(v-1)^{\alpha },N, \Lambda _1}^{\frac{1}{2}}. \end{aligned}$$
We set \(\mathcal{Q}_{N,\beta }(\Lambda _1)=\{\, \text{e}^{{-\frac{1}{2}}\beta (v-1)}\varphi \,|\,\varphi \in \mathcal{P}_{N}(\Lambda _1)\}.\) By (2.8) of [38], we have
$$\begin{aligned} (\phi ,\psi )_{(v-1)^\alpha ,N,\Lambda _1}=(\phi , \psi )_{(v-1)^\alpha ,\Lambda _1}, \quad \quad \forall \phi \in \mathcal{Q}_{m,\beta }(\Lambda _1), \quad \psi \in \mathcal{Q}_{2N-m,\beta }(\Lambda _1). \end{aligned}$$
(4)
In particular (cf. (2.9) of [38]),
$$\begin{aligned} ||\phi ||_{v^{\alpha },N,\Lambda _1}=||\phi ||_{v^{\alpha },\Lambda _1},\quad \quad \quad \forall \phi \in \mathcal{Q}_{N,\beta }(\Lambda _1). \end{aligned}$$
(5)
Moreover, for any \(\phi \in \mathcal{Q}_{N+1,\beta }(\Lambda _1)\) we have (cf. (2.10) of [38])
$$\begin{aligned} ||\phi ||_{(v-1)^{\alpha }, N,\Lambda _1} \le \sqrt{2}\, {d}(N,\alpha )||\phi ||_{(v-1)^{\alpha },\Lambda _1}, \end{aligned}$$
(6)
where \(d(N,\alpha )=\max (\sqrt{1+\frac{\alpha }{N+1}},1).\)

For the pseudospectral method of nonlinear problems with varying coefficients, we need the following result:

Lemma 2.1

For any\(\phi \in \mathcal{Q}_{N,\beta }(\Lambda _1)\), there holds
$$\begin{aligned} \Vert v \phi \Vert _{(v-1)^{\alpha },\Lambda _1}\le c \beta ^{-1} N\Vert \phi \Vert _{(v-1)^{\alpha },\Lambda _1}. \end{aligned}$$
(7)

Proof

Set
$$\begin{aligned} \eta ^{(\alpha ,\beta )}_l(v)=\frac{1}{\sqrt{\gamma _{l}^{(\alpha ,\beta )}}}\tilde{\mathcal{L}}_{l}^{(\alpha ,\beta )}(v-1),\qquad \gamma _{l}^{(\alpha ,\beta )}=\frac{\Gamma (l+\alpha +1)}{\beta ^{\alpha +1}l!}. \end{aligned}$$
As is shown in [24], \(||\tilde{\mathcal{L}}_{l}^{(\alpha ,\beta )}||_{(v-1)^{\alpha },\Lambda _1}^2=\gamma _{l}^{(\alpha ,\beta )}.\) Hence, the set of \(\eta ^{(\alpha ,\beta )}_l(v)\) is the complete normalized \(L^2_{(v-1)^{\alpha }}(\Lambda _1)\)-orthogonal system. Consequently, for any \(\phi \in \mathcal{Q}_{N,\beta }(\Lambda _1)\),
$$\begin{aligned} \phi (v)=\sum _{l=0}^{N}{\hat{\phi }}^{(\alpha ,\beta )}_l\eta ^{(\alpha ,\beta )}_l(v),\qquad {\hat{\phi }}^{(\alpha ,\beta )}_l=(\phi ,\eta ^{(\alpha ,\beta )}_l)_{(v-1)^{\alpha },\Lambda _1}. \end{aligned}$$
We have that (see [24])
$$\begin{aligned} (l+1)\tilde{\mathcal{L}}_{l+1}^{(\alpha ,\beta )}(v-1) +(\beta (v-1)-2l-\alpha -1)\tilde{\mathcal{L}}_{l}^{(\alpha ,\beta )}(v-1) +(l+\alpha )\tilde{\mathcal{L}}_{l-1}^{(\alpha ,\beta )}(v-1)=0,\quad \quad l\ge 1. \end{aligned}$$
By the above formulation,
$$\begin{aligned} v \phi (v)&=\displaystyle \sum _{l=0}^{N}{\hat{\phi }}^{(\alpha ,\beta )}_lv\eta ^{(\alpha ,\beta )}_l(v)=\displaystyle \sum _{l=0}^{N}{\hat{\phi }}^{(\alpha ,\beta )}_l\frac{v}{\sqrt{\gamma _{l}^{(\alpha ,\beta )}}}\tilde{\mathcal{L}}_{l}^{(\alpha ,\beta )}(v-1)\\&=\dfrac{1}{\beta }\displaystyle \sum _{l=0}^{N}{\hat{\phi }}^{(\alpha ,\beta )}_l\big ((2l+\alpha +\beta +1)\eta ^{(\alpha ,\beta )}_l(v)-\sqrt{(l+1)(l+\alpha +1)}\eta ^{(\alpha ,\beta )}_{l+1}(v) -\sqrt{l(l+1)}\eta ^{(\alpha ,\beta )}_{l-1}(v)\big )\\&=\dfrac{1}{\beta }\big ( ((\alpha +1){\hat{\phi }}^{(\alpha ,\beta )}_0-\sqrt{2}{\hat{\phi }}^{(\alpha ,\beta )}_1)\eta ^{(\alpha ,\beta )}_0(v)\\&\quad +\displaystyle \sum _{l=1}^{N-1}((2l+\alpha +\beta +1){\hat{\phi }}^{(\alpha ,\beta )}_l-\sqrt{l(l+\alpha )}{\hat{\phi }}^{(\alpha ,\beta )}_{l-1} -\sqrt{(l+1)(l+2)}{\hat{\phi }}^{(\alpha ,\beta )}_{l+1})\eta ^{(\alpha ,\beta )}_l(v)\\&\quad +((2N+\alpha +\beta +1){\hat{\phi }}^{(\alpha ,\beta )}_N-\sqrt{N(N+\alpha )}{\hat{\phi }}^{(\alpha ,\beta )}_{N-1})\eta ^{(\alpha ,\beta )}_N(v) -\sqrt{(N+1)(N+1+\alpha )}{\hat{\phi }}^{(\alpha ,\beta )}_{N}\eta ^{(\alpha ,\beta )}_{N+1}(v)\big ) \end{aligned}$$
and so
$$\begin{aligned} \Vert v\phi \Vert ^2_{(v-1)^{\alpha },\Lambda _1}&=\dfrac{1}{\beta ^2}\big ( ((\alpha +1){\hat{\phi }}^{(\alpha ,\beta )}_0-\sqrt{2}{\hat{\phi }}^{(\alpha ,\beta )}_1)^2\\&\quad +\displaystyle \sum _{l=1}^{N-1}((2l+\alpha +1){\hat{\phi }}^{(\alpha ,\beta )}_l-\sqrt{l(l+\alpha )}{\hat{\phi }}^{(\alpha ,\beta )}_{l-1} -\sqrt{(l+1)(l+2)}{\hat{\phi }}^{(\alpha ,\beta )}_{l+1})^2\\&\quad +((2N+\alpha +\beta +1){\hat{\phi }}^{(\alpha ,\beta )}_N-\sqrt{N(N+\alpha )}{\hat{\phi }}^{(\alpha ,\beta )}_{N-1})^2 +{(N+1)(N+1+\alpha )}({\hat{\phi }}^{(\alpha ,\beta )}_{N})^2\big )\\&\le \dfrac{6}{\beta ^2}(N+1)(N+\alpha +\beta +1)\Vert \phi \Vert ^2_{(v-1)^{\alpha },\Lambda _1}\le c\beta ^{-2}N^2\Vert \phi \Vert ^2_{(v-1)^{\alpha },\Lambda _1}. \end{aligned}$$
For any \(u\in C[1,\infty )\), the new generalized Laguerre–Gauss–Radau interpolation \(\tilde{\mathcal{I}}_{R, N, \alpha , \beta , \Lambda _1}u \in \mathcal{Q}_{N,\beta }(\Lambda _1)\) is determined by
$$\begin{aligned} \tilde{\mathcal{I}}_{R, N, \alpha , \beta ,\Lambda _1}u({{\tilde{\xi }}}^{(\alpha ,\beta )}_{R,N,\Lambda _1,j}) =u({{\tilde{\xi }}}^{(\alpha ,\beta )}_{R,N,\Lambda _1,j}),\quad 0\le j\le N. \end{aligned}$$
We have that if \(u\in C[1,\infty )\), \(\partial _v^r(\text{e}^{\frac{1}{2} \beta (v-1)}u) \in L^2_{\omega _{r+\alpha , \beta }^1}(\Lambda _1)\cap L^2_{\omega _{r+\alpha -1, \beta }^1}(\Lambda _1),\) integers \(r\ge 1\) and \(r> \alpha +1,\) then (cf. [38])
$$\begin{aligned} ||\tilde{\mathcal{I}}_{R, N, \alpha , \beta , \Lambda _1 }u-u||_{(v-1)^\alpha ,\Lambda _1}&\le c(\beta N)^{\frac{1-r}{2}} (\beta ^{-1}||\partial _v^r(\text{e}^{\frac{1}{2} \beta (v-1)}u) ||_{\omega ^1_{r+\alpha -1,\beta }, \Lambda _1}\\&\quad +(1+\beta ^{-\frac{1}{2}})(\ln N)^{\frac{1}{2}} ||\partial _v^r(\text{e}^{\frac{1}{2} \beta (v-1)}u) ||_{\omega ^1_{r+\alpha ,\beta },\Lambda _1}). \end{aligned}$$
(8)
In particular, if \(|\alpha |< 1\), then the above result holds for any integer \(r\ge 1.\)

Remark 2.1

If \(u \in H^r_{(v-1)^{\alpha +r}}(\Lambda _1)\), then the norm \(||\partial _v^r(\text{e}^{{\frac{1}{2}}\beta (v-1)}u)||_{\omega ^1_{r+\alpha ,\beta },\Lambda _1}\) is finite.

We now consider the interpolation on the subdomain \(\Lambda _2\). The space \(H_{\chi }^{r}(\Lambda _2)\) is defined as usual, with the inner product \((u,w)_{r,\chi ,\Lambda _2}\), the semi-norm \(|u|_{r,\chi ,\Lambda _2}\) and the norm \(||u||_{r,\chi ,\Lambda _2}\). Especially, \(H_{\chi }^0(\Lambda _2)=L_{\chi }^2(\Lambda _2)\), with the inner product \((u,w)_{\chi ,\Lambda _2}\) and the norm \(||u||_{\chi ,\Lambda _2}\).

Let \({\tilde{\xi }}^{(\alpha ,\beta )}_{R,N, \Lambda _2, j}, \, 0 \le j \le N,\) be the zeros of polynomial \(-(v+1)\tilde{\mathcal{L}}_{N}^{(\alpha +1,\beta )}(-v-1)\), which are arranged in descending order. Denote by \({\tilde{\omega }}^{(\alpha ,\beta )}_{R,N,\Lambda _2, j},\, 0 \le j \le N,\) the corresponding Christoffel numbers. For \(v\in \Lambda _2\), we introduce the discrete inner product and norm as follows (cf. [38]):
$$\begin{aligned} (u,w)_{(-v-1)^{\alpha }, N,\Lambda _2}=\sum \limits _{j=0}^{N}u({\tilde{\xi }}^{(\alpha ,\beta )}_{R,N,\Lambda _2,j}) w({\tilde{\xi }}^{(\alpha ,\beta )}_{R,N,\Lambda _2,j}){\tilde{\omega }}^{(\alpha ,\beta )}_{R,N,\Lambda _2,j}, \quad ||u||_{(-v-1)^{\alpha },N, \Lambda _2}=(u,u)_{(-v-1)^{\alpha },N, \Lambda _2}^{\frac{1}{2}}. \end{aligned}$$
We set \(\mathcal{Q}_{N,\beta }(\Lambda _2)=\{\, \text{e}^{{\frac{1}{2}}\beta (v+1)}\varphi \,|\,\varphi \in \mathcal{P}_{N}(\Lambda _2)\}.\) Like (4)–(7), we have that (cf. [38])
$$\begin{aligned} (\phi ,\psi )_{(-v-1)^{\alpha }, N,\Lambda _2}= \,& {} (\phi ,\psi )_{(-v-1)^{\alpha },\Lambda _2},\quad \forall \phi \in \mathcal{Q}_{m,\beta }(\Lambda _2), \psi \in \mathcal{Q}_{2N-m,\beta }(\Lambda _2),\quad 0\le m\le 2N, \end{aligned}$$
(9)
$$\begin{aligned} ||\phi ||_{(-v-1)^{\alpha },N,\Lambda _2}= & {} ||\phi ||_{(-v-1)^{\alpha },\Lambda _2},\quad \forall \phi \in \mathcal{Q}_{N,\beta }(\Lambda _2),\end{aligned}$$
(10)
$$\begin{aligned} ||\phi ||_{(-v-1)^{\alpha }, N,\Lambda _2}\le & {} \sqrt{2}d(N,\alpha )||\phi ||_{(-v-1)^{\alpha },\Lambda _2},\quad \forall \phi \in \mathcal{Q}_{N+1,\beta }(\Lambda _2), \end{aligned}$$
(11)
$$\begin{aligned} ||v\phi ||_{(-v-1)^{\alpha }, \Lambda _2}\le & {} c\beta ^{-1}N||\phi ||_{(-v-1)^{\alpha },\Lambda _2},\quad \forall \phi \in \mathcal{Q}_{N,\beta }(\Lambda _2). \end{aligned}$$
(12)
For any \(u\in C(-\infty ,-1]\), the generalized Laguerre–Gauss–Radau interpolation \(\tilde{\mathcal{I}}_{R, N, \alpha , \beta , \Lambda _2}u \in \mathcal{Q}_{N,\beta }(\Lambda _2)\) is determined by
$$\begin{aligned} \tilde{\mathcal{I}}_{R, N, \alpha , \beta ,\Lambda _2}u({{\tilde{\xi }}}^{(\alpha ,\beta )}_{R,N,\Lambda _2,j}) =u({{\tilde{\xi }}}^{(\alpha ,\beta )}_{R,N,\Lambda _2,j}),\quad \qquad 0\le j\le N. \end{aligned}$$
Let \({\omega^2} _{\alpha ,\beta }(v)={\omega^1} _{\alpha ,\beta }(-v)\). If \(u\in C(-\infty ,-1]\), \(\partial _v^r(\text{e}^{-\frac{1}{2} \beta {(v+1)}}u) \in L^2_{\omega ^2_{r+\alpha ,\beta }}(\Lambda _2)\cap L^2_{\omega ^2_{r+\alpha -1,\beta }}(\Lambda _2)\), integers \(r\ge 1\) and \(r>\alpha +1,\) then (cf. [38])
$$\begin{aligned} ||\tilde{\mathcal{I}}_{R, N, \alpha , \beta , \Lambda _2 }u-u||_{(-v-1)^\alpha ,\Lambda _2}&\le c(\beta N)^{\frac{1-r}{2}} (\beta ^{-1}||\partial _v^r(\text{e}^{-\frac{1}{2} \beta (v+1)}u) ||_{\omega ^2_{r+\alpha -1,\beta }, \Lambda _2}\\&\quad +(1+\beta ^{-\frac{1}{2}})(\ln N)^{\frac{1}{2}} ||\partial _v^r(\text{e}^{-\frac{1}{2} \beta (v+1)}u) ||_{\omega ^2_{r+\alpha ,\beta },\Lambda _2}). \end{aligned}$$
(13)
In particular, if \(|\alpha |< 1\), then the above result holds for any integer \(r\ge 1.\)

Remark 2.2

If \(u\in L^2_{(-v-1)^{r+\alpha }}(\Lambda _2)\), then \(||\partial _v^r(\text{e}^{-{\frac{1}{2}}\beta (v+1)}u)||_{{\omega }^2_{\alpha +r,\beta },\Lambda _2}\) is finite.

2.2 Legendre–Gauss–Lobatto Interpolation

We now consider the Legendre–Gauss–Lobatto interpolation on the interval I. For integers \(r\ge 0\), we define the space \(H^{r}(I)\) and its norm \(||u||_{r,I}\) in the usual way. The inner product and norm of \(L^2(I)\) are denoted by \((u,w)_{I}\) and \(||u||_{I},\) respectively.

The Legendre polynomial of degree l is defined by
$$\begin{aligned} L_{l}(v)=\frac{(-1)^{l}}{2^{l}l!}\partial _v^{l} (1-v^{2})^{l}. \end{aligned}$$
The set of Legendre polynomials is a complete \(L^{2}(I)\)-orthogonal system.
For integers \(M\ge 0,\)\(\mathcal{P}_{M}(I)\) stands for the set of all polynomials of degree at most M. Let \(\zeta _{L,M,I,k}\) be the roots of polynomial \((1-v^2) \partial _vL_M(v), 0 \le k\le M\), which are arranged in ascending order. The corresponding Christoffel numbers are denoted by \(\rho _{L,M,I,k},0\le k\le M.\) We also introduce the discrete inner product and norm as
$$\begin{aligned} (u,w)_{M,I}=\sum \limits _{k=0}^{M}u(\zeta _{L,M,I,k})w(\zeta _{L,M,I,k})\rho _{L,M,I,k}, \qquad \quad ||u||_{M,I}=(u,u)_{M,I}^{\frac{1}{2}}. \end{aligned}$$
We have that (cf. [12])
$$\begin{aligned} (\phi ,\psi )_{M,I}= & {} (\phi ,\psi )_{I}, \quad \quad \quad \qquad \forall \, \phi \psi \in \mathcal{P}_{2M-1}(I), \end{aligned}$$
(14)
$$\begin{aligned} ||\phi ||_{I}\le ||\phi ||_{M,I}\le & {} \sqrt{2+\frac{1}{M}}||\phi ||_{I}, \qquad \quad \forall \phi \in \mathcal{P}_{M}(I). \end{aligned}$$
(15)
Like (7), for \(\phi \in \mathcal{P}_{M}(I)\),
$$\begin{aligned} ||v\phi ||_{I}\le c M||\phi ||_{I}. \end{aligned}$$
(16)
For any \(u\in C[-1,1]\), the Legendre–Gauss–Lobatto interpolation \(\mathcal{I}_{L,M,I}u\in \mathcal{P}_M(I)\) is determined by
$$\begin{aligned} \mathcal{I}_{L,M,I}u(\zeta _{L,M,I,k})=u(\zeta _{L,M,I,k}), \quad \qquad \qquad 0\le k\le M.\quad \end{aligned}$$
According to (2.10) of [27], we have that if \(u\in C[-1,1], \, (1-v^{2})^{\frac{r-1}{2}}\partial _v^{r}u\in L^2(I)\) and integers \(r\ge 1,\) then
$$\begin{aligned} || \mathcal{I}_{L,M,I}u-u||_{I} \le cM^{-r}||(1-v^2)^{\frac{r-1}{2}} \partial _v^{r}{u}||_{I}. \end{aligned}$$
(17)

2.3 Composite Generalized Laguerre–Legendre Interpolation on the Whole Line

We are now in position of studying the composite Laguerre–Legendre interpolation on the whole line \(\mathbb {R}\). The space \(L_{\chi }^{2}(\mathbb {R})\) is defined as usual, with the inner product \((u,w)_{\chi ,\mathbb {R}}\) and the norm \(||u||_{\chi ,\mathbb {R}}\). Further, let
$$\begin{aligned} H^1_{1,1+v^2}(\mathbb {R})=\{\,u\,|\,u \in L_{v^2+1}^2({\mathbb {R}}) \, \mathrm{and}\, \partial _vu\in L^2(\mathbb {R})\}, \end{aligned}$$
and
$$\begin{aligned} \mathcal{Q}_{M,N,\beta }(\mathbb {R})=H^1_{1,1+v^2}({\mathbb {R}})\cap \{\,\phi \,|\,\phi \,|_{I}\in \mathcal{P}_{M}(I),\,\phi \,|_{\Lambda _1}\in \mathcal{Q}_{N,\beta }(\Lambda _1)\,\mathrm{and} \,\phi \,|_{\Lambda _2} \in \mathcal{Q}_{N,\beta }(\Lambda _2)\}. \end{aligned}$$
We introduce the discrete inner product and norm as
$$\begin{aligned} (u,w)_{\omega ^{(\alpha )},M,N,\mathbb {R}}= & {} (u,w)_{M,I}+(u,w)_{(v-1)^{\alpha },N,\Lambda _1}+(u,w)_{(-v-1)^{\alpha },N,\Lambda _2},\nonumber \\ ||u||_{\omega ^{(\alpha )},M,N,\mathbb {R}}= & {} (u,u)_{\omega ^{(\alpha )},M,N,\mathbb {R}}^{\frac{1}{2}}, \end{aligned}$$
(18)
with
$$\begin{aligned} \omega ^{(\alpha )}(v)=\left\{ \begin{array}{ll}1,\qquad &{}\quad v\in I,\\ (|v|-1)^{\alpha }, &{}\quad v\in \mathbb {R}\setminus {\bar{I}}.\end{array}\right. \end{aligned}$$
Denote
$$\begin{aligned} {\bar{d}}(M,N,\alpha )=\sqrt{2}\left (\sqrt{2+\frac{1}{M}}+\sqrt{1+\dfrac{\alpha }{N+1}}\,\right ). \end{aligned}$$
By virtue of (4)–(7), (9)–(12) and (14)–(16), we have that
$$\begin{aligned} (\phi ,\psi )_{\omega ^{(\alpha )},M, N,\mathbb {R}}= & {} (\phi ,\psi )_{ \omega ^{(\alpha )},\mathbb {R}},\quad \quad \forall \phi \in \mathcal{Q}_{m,n,\beta } (\mathbb {R}),\quad \psi \in \mathcal{Q}_{2M-1-m,2N-n,\beta }(\mathbb {R}),\end{aligned}$$
(19)
$$\begin{aligned} \Vert \phi \Vert _{\omega ^{(\alpha )}, M, N,\mathbb {R}}= & {} \Vert \phi \Vert _{\omega ^{(\alpha )},\mathbb {R}},\quad \quad \forall \phi \ \in \mathcal{Q}_{M-1,N,\beta }(\mathbb {R}),\end{aligned}$$
(20)
$$\begin{aligned} ||\phi ||_{\omega ^{(\alpha )},\mathbb {R}}\le ||\phi ||_{\omega ^{(\alpha )},M, N,\mathbb {R}}\le & {} \left(1+\sqrt{2+\frac{1}{M}}\right)||\phi ||_{\omega ^{(\alpha )},\mathbb {R}}, \quad \forall \phi \in \mathcal{Q}_{M,N,\beta }(\mathbb {R}), \end{aligned}$$
(21)
$$\begin{aligned} ||\phi ||_{\omega ^{(\alpha )}, M, N,\mathbb {R}}\le & {} {\bar{d}}(M,N,\alpha )||\phi ||_{\omega ^{(\alpha )},\mathbb {R}},\quad \forall \phi \in \mathcal{Q}_{M,N+1,\beta }(\mathbb {R}), \end{aligned}$$
(22)
$$\begin{aligned} ||v\phi ||_{\omega ^{(\alpha )},\mathbb {R}}\le & {} c(\beta ^{-1}N+{M})||\phi ||_{\omega ^{(\alpha )},\mathbb {R}},\qquad \forall \phi \in \mathcal{Q}_{M, N,\beta }(\mathbb {R}). \end{aligned}$$
(23)
The composite Legendre–Laguerre interpolation \(I_{M,N,{\alpha },\beta ,\mathbb {R}}u(v)\in \mathcal{Q}_{M,N,\beta }(\mathbb {R})\) is defined by
$$\begin{aligned} I_{M,N,{\alpha },\beta ,\mathbb {R}}u(v)=\left\{ \begin{array}{ll} \mathcal{I}_{L,M,I}u(v), \qquad &{}v\in \{\zeta _{L,M,I,k}, \,0\le k \le M\},\\ \tilde{\mathcal{I}}_{R, N, \alpha , \beta , \Lambda _j }u(v), &{} v\in \{{{\tilde{\xi }}}^{(\alpha ,\beta )}_{R,N,\Lambda _j,k},\,0\le k\le N,\, j=1,2\}.\end{array}\right. \end{aligned}$$
Let \({\bar{\omega }}_{\alpha ,\beta }|_{\Lambda _j}={\omega }^j_{\alpha ,\beta }, j=1,2.\) We use the following notation:
$$\begin{aligned} {\mathbb {C}}^{q,r}_{N,\alpha ,\beta ,\mathbb {R}}(u) =\quad&\beta ^{-2}||\partial _v^r(\text{e}^{\frac{1}{2} \beta (|v|-1)}u) ||^2_{\bar{\omega }_{r+\alpha -1,\beta }, \,\Lambda _1\cup \Lambda _2} +||(1-v^2)^{\frac{q-1}{2}}\partial _v^{q}{u}||^2_{I}\\&+(1+\beta ^{-1})\ln N ||\partial _v^r(\text{e}^{\frac{1}{2} \beta (|v|-1)}u) ||^2_{\bar{\omega }_{r+\alpha ,\beta }, \,\Lambda _1\cup \Lambda _2}. \end{aligned}$$
Using (8), (13) and (17), we obtain that
$$\begin{aligned} ||I_{M,N,{\alpha },\beta ,\mathbb {R}}u-u||^2_{\omega ^{(\alpha )},\mathbb {R}}&\le c\big ((\beta N)^{ {1-r}}+M^{-2q}\big ){\mathbb {C}}^{q,r}_{N,\alpha , \beta , \mathbb {R}}(u). \end{aligned}$$
(24)
In particular, if \(|\alpha |< 1\), then the above result holds for any integer \(r\ge 1.\)
In numerical analysis of the composite pseudospectral method with multi-domain decomposition for the Fokker–Planck equation on the whole line, we need a composite quasi-orthogonal approximation. Thereby, we study a non-standard projection. To do this, for \(j=1,2\), let (cf. [38])
$$\begin{aligned} { H}^1_{\chi _1,\chi _0}(\Lambda _j)= & {} \{\,u\,|\, u\in L^2_{\chi _0}(\Lambda _j)\,\mathrm{and}\, \partial _v u\in L^2_{\chi _1}(\Lambda _j)\},\\ {\bar{H}}^1_{\chi _1,\chi _0}(\Lambda _j)= & {} \{\,u\,|\, u\in { H}^1_{\chi _1,\chi _0}(\Lambda _j) {\,\mathrm {and}\,}u(v^j_b)=0\}, \end{aligned}$$
where \(v^1_b=1, v^2_b=-1.\) Set \(\bar{\mathcal{P}}_N(\Lambda _j)=\{\phi \,|\,\phi \in \mathcal{P}_N(\Lambda _j),\,\phi (v^j_b)=0\}.\) Let \(\xi _\beta (v)=(v^2+1)\cdot\text{e}^{-\beta (|v|-1)}\). The auxiliary orthogonal projections \({\bar{P}}^1_{N,\beta ,\Lambda _j}:\bar{H}^1_{\omega _{0,\beta },\xi _\beta }(\Lambda _j)\rightarrow \bar{\mathcal{P}}_N(\Lambda _j)\) are defined by
$$\begin{aligned} (\partial _v({\bar{P}}^1_{N,\beta ,\Lambda _j}u-u),\partial _v \phi )_{\bar{\omega }_{0,\beta }, \Lambda _j}+({\bar{P}}^1_{N,\beta ,\Lambda _j}u-u,\phi )_{\xi _\beta , \Lambda _j}=0,\quad \quad \forall \phi \in \bar{\mathcal{P}}_N(\Lambda _j),\quad j=1,2. \end{aligned}$$
For any \(u\in { H}^1_{\chi _1,\chi _0}(\Lambda _j),\,{\tilde{u}}(v)=u(v)-u(v^j_b)\text{e}^{-\frac{1}{2} \beta (|v|-1)}\). The quasi-orthogonal projections on \(\Lambda _j\) are defined by
$$\begin{aligned} P^1_{N,\beta ,\Lambda _j}u=\text{e}^{-\frac{ 1}{2}\beta (|v|-1)}({\bar{P}}^1_{N,\beta ,\Lambda _j} (\text{e}^{\frac{1}{2}\beta (|v|-1) }\tilde{u})+u(v^j_b))\in \mathcal{Q}_{N, \beta }(\Lambda _j),\qquad j=1,2. \end{aligned}$$
By Lemma 2.1 and (2.13) of [26], we have that if \(u \in H^1_{1, v^{2}+ 1}(\Lambda _j), \,\partial _v^{r+1}(\text{e}^{{\frac{1}{2}}\beta (|v|-1)}u)\in L^2_{\omega ^j_{r+1,\beta }}(\Lambda _j)\) and integers \(r\ge 1\), then
$$\begin{aligned} &||\partial _v(P^1_{N,\beta ,\Lambda _j}u-u)||_{\Lambda _j}+||P^1_{N,\beta ,\Lambda }u-u||_{v^{2}+1,\,\Lambda _j}\\& \le c(\beta +\beta ^{-2})(\beta N)^{\frac{1-r}{2}}||\partial _v^{r+1}(\text{e}^{{\frac{1}{2}}\beta (|v|-1)} u)||_{\omega ^{j}_{r+1,\beta },\Lambda _j}, \quad j=1,2. \end{aligned}$$
(25)
Next, we consider the quasi-orthogonal approximation defined on of \(H^1(I)\). Let \(\mathcal{P}^0_{M}(I)=H_0^1(I)\cap \mathcal{P}_{M}(I)\). The auxiliary orthogonal projection \({P}_{M,I}^{1,0}:H_0^1(I)\rightarrow \mathcal{P}^0_{M}(I)\) is defined by
$$\begin{aligned} (\partial _v({P}_{M,I}^{1,0}u-u),\partial _v\phi )_{I}=0, \qquad \forall \phi \in \mathcal{P}^0_{M}(I). \end{aligned}$$
For any \(u \in H^1(I)\), we set \(\tilde{u}(v)=u(v)-\frac{1}{2} u(1)(v+1)+\frac{1}{2} u(-1)(v-1) \in H_0^1(I).\) Then, we define the quasi-orthogonal projection \(_{*}P_{M,I}^1\) as
$$\begin{aligned} _{*}P_{M,I}^1 u(v)=P_{M,I}^{1,0}\tilde{u}(v)+\frac{1}{2} u(1)(v+1)-\frac{1}{2} u(-1)(v-1). \end{aligned}$$
Thanks to Lemma 2.4 of [25], we have that if \(\partial _v^s u \in L^2(I), \,(1-v^2)^{\frac{q-1}{2}} \partial _v^{q}u\in L^2(I)\) and integers \(q\ge 1\), then
$$\begin{aligned} ||\partial _v^s (_{*}P^1_{M,I}u-u)||_{I} \le c M^{s-q}||(1-v^2)^{\frac{q-1}{2}} \partial _v^{q}u||_{I},\qquad s=0,1.\end{aligned}$$
(26)
The above formulation implies that if \(u\in H^1_{1, v^{2}+ 1}(I)\),
$$\begin{aligned} ||\partial _v(_{*}P^1_{M,I}u-u)||_{I}+||_{*}P^1_{M,I}u-u||_{v^{2}+ 1, I} \le c M^{1-q}||(1-v^2)^{\frac{q-1}{2}} \partial _v^{q}u||_{I}. \end{aligned}$$
(27)
To analyze the numerical errors, we need a non-standard projection \({P}^1_{M,N,\beta ,\mathbb {R}}:\)\(H^1_{1, v^{2}+ 1}(\mathbb {R})\rightarrow Q_{M,N,\beta }(\mathbb {R}),\) which is defined by
$$\begin{aligned} (\partial _v({P}^1_{M,N,\beta ,\mathbb {R}}u-u),\partial _v \phi )_{\mathbb {R}}+({P}^1_{M,N,\beta ,\mathbb {R}}u-u,\phi )_{v^2+1, \mathbb {R}}=0,\quad \quad \forall \phi \in \mathcal{Q}_{M,N,\beta }(\mathbb {R}). \end{aligned}$$
(28)
By the projection theorem,
$$\begin{aligned} ||\partial _v(P^1_{M,N,\beta ,\mathbb {R}}u-u)||_{\mathbb {R}} +||P^1_{M,N,\beta ,\mathbb {R}}u-u||_{v^{2}+1,\mathbb {R}} \le ||\partial _v(\phi -u)||_{\mathbb {R}}+||\phi -u||_{v^{2}+1,\mathbb {R}}, \quad \forall \phi \in Q_{M,N,\beta }(\mathbb {R}). \end{aligned}$$
(29)
Take \(\phi =P^1_{N,\beta ,\Lambda _j}u\) on \(\Lambda _j,j=1,2\), and \(\phi ={}_{*}P^1_{M,I}u\) on I. By the definition of \(P^1_{N,\beta ,\Lambda _j}\) and \({}_{*}P^1_{M,I}\), we conclude that \(\phi \in Q_{M,N,\beta }(\mathbb {R})\). Therefore, with the aid of (25) and (27), we verify that
$$\begin{aligned} &||\partial _v(P^1_{M,N,\beta ,{ \mathbb {R}}}u-u)||_{\mathbb {R}} +||P^1_{M,N,\beta ,\mathbb {R}}u-u||_{v^{2}+1,\mathbb {R}}\\& \le c(\beta +\beta ^{-2})(\beta N)^{\frac{1-r}{2}}||\partial _v^{r+1}(\text{e}^{{\frac{1}{2}}\beta (|v|-1)} u)||_{\bar{\omega }_{r+1,\beta },\Lambda _1\cup \Lambda _2}+c M^{1-q}||(1-v^2)^{\frac{q-1}{2}} \partial _v^{q}u||_{I}\\& \le c\big ((\beta N)^{\frac{1-r}{2}} +M^{1-q}\big ){\mathscr {B}}^{q,r}_{\beta ,\mathbb {R}}(u), \end{aligned}$$
(30)
where \({\mathscr {B}}^{q,r}_{\beta ,\mathbb {R}}(u)=(\beta +\beta ^{-2})||\partial _v^{r+1}(\text{e}^{{\frac{1}{2}}\beta (|v|-1)}u)||_{\bar{\omega }_{r+1,\beta },\Lambda _1\cup \Lambda _2}+||(1-v^2)^{\frac{q-1}{2}}\partial _v^{q}u||_{I}\) with \(q,r>1.\)

In the end of this section, we introduce some inverse inequalities which will be used in the sequel (cf. [12, 37]).

Proposition 2.1

For any\(\phi \in \mathcal{Q}_{N,\beta }(\Lambda _j)\) and\(1\le p\le q \le \infty\), there holds
$$\begin{aligned} \Vert \phi \text{e}^{(\frac{\beta }{2} -\frac{\beta }{q})(|v|-1)}\Vert _{L^{q}(\Lambda _j)} \le c(\beta N)^{(\frac{1}{p} - \frac{1}{q})}\Vert \phi \text{e}^{(\frac{\beta }{2} -\frac{\beta }{p})(|v|-1)}\Vert _{L^{p}(\Lambda _j)}, \quad j=1,2. \end{aligned}$$
(31)

Proposition 2.2

For any\(\phi \in \mathcal{Q}_{M,N,\beta }(\mathbb {R})\), there hold
$$\begin{aligned} \Vert \partial _v\phi \Vert _{\mathbb {R}}\le \,& {} c (\beta ^{-1}N+{M^2})||\phi ||_{\mathbb {R}},\end{aligned}$$
(32)
$$\begin{aligned} \Vert v \phi \Vert _{\mathbb {R}}\le\, & {} c (\beta ^{-1} N+M)\Vert \phi \Vert _{\mathbb {R}}. \end{aligned}$$
(33)

Remark 2.3

By Theorem 2.7 of [12] and Proposition 2.1, for any \(\phi \in \mathcal{Q}_{M,N,\beta }(\mathbb {R})\) and \(q\ge 1\), there holds
$$\begin{aligned} \Vert \phi \Vert _{L^{2q}(\mathbb {R})}\le c\Big ((\beta N)^{(\frac{1}{2} - \frac{1}{2q})}+M^{(1 - \frac{1}{q})}\Big )\Vert \phi \Vert _{\mathbb {R}}. \end{aligned}$$
(34)

3 Composite Pseudospectral Scheme for Fokker–Planck Equation

In this section, we propose the composite pseudospectral scheme for the nonlinear Fokker–Planck equation on the whole line, together with the convergence analysis. We also describe the implementation and present some numerical results.

3.1 Pseudospectral Scheme

A weak formulation of (1) is to seek \(W \in L^\infty (0, T; L^2(\mathbb {R})) \cap L^2(0, T; H^1_{1,v^2+1}(\mathbb {R}))\), such that
$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _tW(t),u)_{\mathbb {R}}+ (v W(t),\partial _vu)_{\mathbb {R}} +k(v W^2(t),\partial _vu)_{\mathbb {R}}+(\partial _vW(t),\partial _vu)_{\mathbb {R}} =0,\\\quad \forall \, u \in H^1_{1, v^{2}+1}(\mathbb {R}),\quad 0<t\le T,\\ W(0)=W_{0}. \end{array} \right. \end{aligned}$$
(35)
We now design the composite generalized Laguerre–Legendre pseudospectral scheme for (35). It is to find \(w_{M,N}(t)\in \mathcal{Q }_{M,N,\beta }(\mathbb {R})\) for all \(0\le t\le T\), such that
$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _tw_{M,N}(t),\phi )_{M,N,\mathbb {R}}+ (v w_{M,N}(t), \partial _v\phi )_{M,N,\mathbb {R}} +\,k(v w^2_{M,N}(t),\partial _v\phi )_{M,N,\mathbb {R}}\\ +(\partial _vw_{M,N}(t),\partial _v\phi )_{M,N,\mathbb {R}} =0,\qquad \qquad \forall \phi \in \mathcal{Q}_{M,N,\beta }(\mathbb {R}),\quad 0<t\le T,\\ w_{N}(0)=I_{M,N,0,\beta ,\mathbb {R}}W_0 \,\mathrm{or\,}P^1_{M,N,\beta ,\mathbb {R}}W_{0}. \end{array}\right. \end{aligned}$$
(36)
Thanks to (19), the above problem is equivalent to
$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _tw_{M,N}(t),\phi )_{M,N,\mathbb {R}}+ (v w_{M,N}(t),\partial _v\phi )_{M,N,\mathbb {R}} +\,k(v w^2_{M,N}(t),\partial _v\phi )_{M,N,\mathbb {R}}\\ +(\partial _vw_{M,N}(t),\partial _v\phi )_{\mathbb {R}} =0,\qquad \qquad \forall \phi \in \mathcal{Q}_{M,N,\beta }(\mathbb {R}),\quad 0<t\le T,\\ w_{M,N}(0)=I_{M,N,0,\beta ,\mathbb {R}}W_0 \,\mathrm{or\,}P^1_{M,N,\beta ,\mathbb {R}}W_{0}. \end{array} \right. \end{aligned}$$
(37)

3.2 Numerical Results

In this section, we describe the implementation of the pseudospectral scheme (37) with a nonhomogeneous term f(vt), and present some numerical results. We use the Crank–Nicolson discretization in time t, with the mesh size \(\tau\).

For simplicity, we use the notation
$$\begin{aligned} a_{M,N,\mathbb {R}}(z,\phi )=(v z,\partial _v\phi )_{M,N,\mathbb {R}}+(\partial _vz,\partial _v\phi )_{\mathbb {R}}. \end{aligned}$$
The fully discrete scheme of (37) is as follows: for \(t= 0,\tau ,\ldots ,T-\tau ,\)
$$\begin{aligned} \left\{ \begin{array}{ll} \dfrac{1}{\tau } ( w_{M,N}(t+\tau )-w_{M,N}(t),\phi )_{M,N,\mathbb {R}} +\dfrac{1}{2}a_{M,N,\mathbb {R}}(w_{M,N}(t+\tau )+w_{M,N}(t),\phi )\\ =\dfrac{1}{2} (f(t+\tau )+f(t),\phi )_{M,N,{\mathbb {R}}}-\dfrac{1}{2} k\big (v (w^2_{M,N}(t+\tau )+w^2_{M,N}(t)),\partial _v\phi \big )_{M,N,{\mathbb {R}}},\\ w_{M,N}(0)=I_{M,N,0,\beta ,{\mathbb {R}}}W_0 \,\mathrm{or\,}P^1_{M,N,\beta ,{\mathbb {R}}}W_{0}. \end{array} \right. \end{aligned}$$
(38)
Let
$$\begin{aligned} A_{M,N,{\mathbb {R}}}(z,u)= & {} \frac{1}{2}\tau a_{M,N,{\mathbb {R}}}(z,u)+(z,u)_{M,N,{\mathbb {R}}},\\ {{\bar{A}}}_{M,N,{\mathbb {R}}}(z,u)= & {} -\frac{1}{2}\tau a_{M,N,{\mathbb {R}}}(z,u)+(z,u)_{M,N,{\mathbb {R}}}. \end{aligned}$$
Then, at each time step, we need to solve the following nonlinear equation:
$$\begin{aligned} A_{M,N,{\mathbb {R}}}(w_{M,N}(t),\phi ) =&{{\bar{A}}}_{M,N,{\mathbb {R}}}(w_{M,N}(t-\tau ),\phi )+\dfrac{1}{2} \tau (f(t)+f(t-\tau ),\phi )_{M,N,{\mathbb {R}}}\\&-\dfrac{1}{2} k\tau \big (v (w^2_{M,N}(t+\tau )+w^2_{M,N}(t)),\partial _v\phi \big )_{M,N,{\mathbb {R}}},\quad \forall \, \phi \in \mathcal{Q}_{M,N,\beta }({\mathbb {R}}). \end{aligned}$$
(39)
For notational convenience, we denote \(\tilde{\mathcal{L}}_{l}^{(0,\beta )}(v)\) by \(\tilde{\mathcal{L}}_{l}^{(\beta )}(v)\). Let (cf. [25])
$$\begin{aligned} \psi ^{(\beta )}_{1,l}(v)= & {} \tilde{\mathcal{L}}_{l}^{(\beta )}(v-1)-\tilde{\mathcal{L}}_{l+1}^{(\beta )}(v-1),\qquad 0 \le l \le N-1,\\ \psi ^{(\beta )}_{2,l}(v)= & {} \tilde{\mathcal{L}}_{l}^{(\beta )}(-v-1)-\tilde{\mathcal{L}}_{l+1}^{(\beta )}(-v-1), \qquad 0 \le l \le N-1, \end{aligned}$$
and
$$\begin{aligned} G_{l}^{(1,\beta )}(v)=\left\{ \begin{array}{ll} \psi ^{(\beta )}_{1,l}(v),&{}\, v\in \Lambda _1 ,\\ 0, &{}\,\mathrm{otherwise}, \end{array}\right. \qquad \qquad G_{l}^{(2,\beta )}(v)=\left\{ \begin{array}{ll} \psi ^{(\beta )}_{2,l}(v) ,&{}\, v\in \Lambda _2 , \\ 0, &{}\,\mathrm{otherwise}. \end{array}\right. \end{aligned}$$
For \(0\le k\le M-2\), let \(\phi _k(v)=L_k(v)-L_{k+2}(v)\), and we set
$$\begin{aligned} G_{k}(v)=\left\{ \begin{array}{ll} \phi _k(v),&{}\, v\in I,\\ 0, &{}\,\mathrm{otherwise}. \end{array}\right. \end{aligned}$$
Furthermore, we consider the following base functions corresponding to the common boundaries of \(\Lambda _j\,(j=1,2)\) and I, respectively. Let
$$\begin{aligned} G_{+}^{(\beta )}(v)=\left\{ \begin{array}{ll} \frac{1+v}{2},&{} v\in {\bar{I}},\\ \text{e}^{\frac{\beta }{2} (1-v)}, &{} v\in \bar{\Lambda }_1,\\ 0, &{}\,\mathrm{otherwise}, \end{array}\right. \qquad \qquad G_{-}^{(\beta )}(v)=\left\{ \begin{array}{ll} \frac{1-v}{2},&{} v\in {\bar{I}},\\ \text{e}^{\frac{\beta }{2} (1+v)}, &{} v\in \bar{\Lambda }_2,\\ 0, &{}\,\mathrm{otherwise}. \end{array}\right. \end{aligned}$$
The functions  \(G_{l}^{(j,\beta )}(v), j=1,2,\)   \({G}_+^{(\beta )}(v), {G}_-^{(\beta )}(v)\) and \(G_k(v)\) form a basis of   \(\mathcal{Q}_{M,N,\beta }({\mathbb {R}})\).
In actual computation, we expand the numerical solution as
$$\begin{aligned} w_{M,N}(v,t)=\sum \limits _{j=1}^{2}\sum \limits _{l=0}^{N-1}v^{(j)}_{l}(t) G^{(j,\beta )}_{l}(v)+\sum \limits _{k=0}^{M-2}v_{k}(t) G_k(v)+v_+(t){G}_{+}^{(\beta )}(v)+ v_-(t){G}_{-}^{(\beta )}(v). \end{aligned}$$
(40)
We use scheme (38) to solve problem (1) with \(k=1\). Now, we take the test function
$$\begin{aligned} W(v,t)=\frac{1}{4\sqrt{t+1}} \tanh \left(\dfrac{\sqrt{6}}{12}\left(v-\dfrac{5\sqrt{6}}{6}t \right)\right)\text{e}^{-\frac{1}{2} v^2}. \end{aligned}$$
The numerical errors are measured by the discrete \(L^2\)-error norm
$$\begin{aligned} E_{M,N}(t)=||W(t)-w_{M,N}(t)||_{M,N,{\mathbb {R}}}. \end{aligned}$$
In Fig. 1, we plot the errors \(\log _{10}E_{M,N}(t)\) with \(N=2M, t=1\) and \(\beta =7\) vs. M. Clearly, the errors decay fast when M and N increase and \(\tau\) decreases. The above facts coincide very well with the theoretical analysis in Sect. 3.3. In particular, they show the spectral accuracy in space of scheme (37).

In Fig. 2, we plot \(\log _{10}E_{M,N}(t)\) at \(N=2M,t=1\), \(\tau =0.000\,25\) and different values of the parameter \(\beta\) vs. M. It seems that the errors with suitably bigger \(\beta\) are smaller than those with smaller \(\beta\). However, the determination of the parameter \(\beta\) is still an open problem. Roughly speaking, if the exact solution decays faster as v increases, then it is better to take suitably bigger \(\beta\).

In Fig. 3 we plot \(\log _{10}E_{M,N}(t)\) with \(\beta =7, M=11, N=22, \tau = 0.000\,25\) vs. t, which indicates the stability of scheme (37).
Fig. 1

\(L^2\)-errors against M with \(N=2M,t=1\), \(\beta =7\)

Fig. 2

\(L^2\)-errors against M with \(N=2M,t=1\), \(\tau =0.000\,25\)

Fig. 3

The numerical errors at \(t=0\) to \(t=30\)

3.3 Convergence Analysis

We next deal with the convergence of scheme (37). Let \(W_{M,N}=P_{M,N,\beta ,{\mathbb {R}}}^{1}W\). We derive from (35) that for \(0<t\le T,\)
$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _tW_{M,N}(t),\phi )_{M,N,{ \mathbb {R}}}+( v W_{M,N}(t),\partial _v\phi )_{M,N,{\mathbb {R}}}+k(v W^2_{M,N}(t), \partial _v\phi )_{M,N,{\mathbb {R}}} \\ +(\partial _v W_{M,N}(t),\partial _v \phi )_{\mathbb {R}} +\sum \limits _{j=1}^{4}G_{j}(t,\phi )=0,\qquad \qquad \forall \,\phi \in \mathcal{Q}_{M,N,\beta }({\mathbb {R}}),\\ W_{M,N}(0)=P^1_{M,N,\beta ,{\mathbb {R}}}W_{0}, \end{array}\right. \end{aligned}$$
(41)
where
$$\begin{aligned} &G_{1}(t,\phi )=(\partial _tW(t) ,\phi )_{{\mathbb {R}}}-(\partial _tW_{M,N}(t) ,\phi )_{M,N,{\mathbb {R}}},\qquad \\&G_{2}(t,\phi )=(vW(t),\partial _v\phi )_{{\mathbb {R}}}-(v W_{M,N}(t), \partial _v\phi )_{M,N,{\mathbb {R}}},\\&G_{3}(t,\phi )=(\partial _v(W(t)- W_{M,N}(t)),\partial _v \phi )_{{\mathbb {R}}},\quad \\&G_{4}(t,\phi )=k(vW^2(t), \partial _v\phi )_{{\mathbb {R}}}- k(vW^2_{M,N}(t), \partial _v\phi )_{M,N,{\mathbb {R}}}. \end{aligned}$$
Taking \(\widetilde{W}_{M,N}=w_{M,N}-W_{M,N}\) and subtracting (41) from (37), we obtain that for \(0<t\le T,\)
$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _t\widetilde{W}_{M,N}(t),\phi )_{M,N,{\mathbb {R}}} +(\partial _v \widetilde{W}_{M,N}(t),\partial _v \phi )_{\mathbb {R}} =\sum \limits _{j=1}^{6}G_{j}(t,\phi ),\qquad \forall \,\phi \in \mathcal{Q}_{M,N,\beta }({\mathbb {R}}),\\ \widetilde{W}_{M,N}(0)=I_{M,N,0,\beta ,{ \mathbb {R}}}W_{0} -P^1_{M,N,\beta ,{\mathbb {R}}}W_{0}\,\mathrm{or\,}0, \end{array} \right. \end{aligned}$$
(42)
where
$$\begin{aligned} G_{5}(t,\phi )=k(v (W^2_{M,N}(t)-w^2_{M,N}(t)), \partial _v\phi )_{M,N,{\mathbb {R}}}, \quad G_{6}(t,\phi )=-( v \widetilde{W}_{M,N}(t),\partial _v\phi )_{M,N,{\mathbb {R}}}. \end{aligned}$$
Take \(\phi =2\widetilde{W}_{N}(t)\) in (42), we deduce that
$$\begin{aligned} \partial _t||\widetilde{W}_{M,N}(t)||_{M,N,{\mathbb {R}}}^2+2||\partial _v \widetilde{W}_{M,N}(t)||_{{\mathbb {R}}}^2 =2\sum \limits _{j=1}^{6}G_{j}(t,\phi ),\quad 0<t\le T. \end{aligned}$$
(43)
Therefore, it suffices to estimate the terms \(|G_{j}(t,\widetilde{W}_{M,N})|\). First, we use the Cauchy inequality, (22) with \(\alpha =0\) and (30) to verify that for integers \(r\ge 1\),
$$\begin{aligned} |G_{1}(t,\widetilde{W}_{M,N}(t))|&=2|(\partial _tW(t) ,\widetilde{W}_{M,N}(t))_{{\mathbb {R}}}-(\partial _tW_{M,N}(t) ,\widetilde{W}_{M,N}(t))_{M,N,{\mathbb {R}}}|\\&=2|(\partial _tW(t)-P_{M-1,N,\beta ,{\mathbb {R}}}^{1}\partial _tW(t) ,\widetilde{W}_{M,N}(t))_{{\mathbb {R}}}\\&\quad +(P_{M-1,N,\beta ,{\mathbb {R}}}^{1}\partial _tW(t)-\partial _tW_{M,N}(t) ,\widetilde{W}_{M,N}(t))_{M,N,{\mathbb {R}}}|\\&\le c\big ((\beta N)^{{1-r}}+M^{2-2q}\big )\big ({\mathscr {B}}^{q,r}_{\beta ,{\mathbb {R}}}(\partial _tW(t))\big )^2+||\widetilde{W}_{M,N}(t)||_{{\mathbb {R}}}^{2}. \end{aligned}$$
(44)
Similarly,
$$\begin{aligned}& |G_{2}(t,\widetilde{W}_{M,N}(t))+G_{3}(t,\widetilde{W}_{M,N}(t))|\\ & \quad \le c\big ((\beta N)^{{1-r}}+M^{2-2q}\big )\big ({\mathscr {B}}^{q,r}_{\beta ,{\mathbb {R}}}(W(t))\big )^2+\dfrac{1}{8}||\partial _v\widetilde{W}_{M,N}(t)||_{{\mathbb {R}}}^{2}. \end{aligned}$$
(45)
Obviously,
$$\begin{aligned} |G_{4}(t,\widetilde{W}_{M,N}(t))|&\le 2 |(v(W^2(t)- I_{M-1,N,\alpha ,\beta ,{\mathbb {R}}}W^2(t)),\partial _v\widetilde{W}_{M,N}(t))_{{\mathbb {R}}}|\\&\quad +|( v(I_{M-1,N,\alpha ,\beta ,{\mathbb {R}}}W^2(t)- W^2_{M,N}(t)),\partial _v\widetilde{W}_{M,N}(t))_{M,N,{\mathbb {R}}}|. \end{aligned}$$
(46)
According to \(v^2\le 2(|v|-1)^2+2\) and (24) with \(\alpha =0,2\), it is easy to derive that
$$\begin{aligned} &2|(v(W^2(t)- I_{M-1,N,\alpha ,\beta ,{\mathbb {R}}}W^2(t)),\partial _v\widetilde{W}_{M,N}(t))_{{\mathbb {R}}}|\\& \le \dfrac{1}{16} ||\partial _v\widetilde{W}_{M,N}(t)||_{{\mathbb {R}}}^{2}+ c\big ((\beta N)^{ {1-r}}+M^{-2q}\big ) \big ({\mathbb {C}}^{q,r}_{N,0, \beta ,{\mathbb {R}}}(W^2(t))+{\mathbb {C}}^{q,r}_{N, 2, \beta ,{\mathbb {R}}}(W^2(t))\big ). \end{aligned}$$
By (22) with \(\alpha =0\), (34) with \(q=\infty\), (24) with \(\alpha =0,2\) and (30), we have that
$$\begin{aligned} &2|( v(I_{M-1,N,\alpha ,\beta ,{\mathbb {R}}}W^2(t)- W^2_{M,N}(t)),\partial _v\widetilde{W}_{M,N}(t))_{M,N,{ \mathbb {R}}}|\\& =2|( v(I_{M-1,N,\alpha ,\beta ,{ \mathbb {R}}}W(t)+ W_{M,N}(t))(I_{M-1,N,\alpha ,\beta ,{ \mathbb {R}}}W(t)- W_{M,N}(t)),\partial _v\widetilde{W}_{M,N}(t))_{M,N,{ \mathbb {R}}}|\\& \le c(\Vert W(t)\Vert ^2_{\infty }+\Vert W_{M,N}(t)\Vert ^2_{\infty })\Vert v(I_{M-1,N,\alpha ,\beta ,{ \mathbb {R}}}W(t)- W_{M,N}(t))\Vert ^2_{M,N,{ \mathbb {R}}}+\frac{1}{16} \Vert \partial _v\widetilde{W}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}\\& \le c\big (\Vert W(t)\Vert ^2_{\infty }+(\beta N+M^2)\Vert W_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}\big )\Vert v(I_{M-1,N,\alpha ,\beta ,{ \mathbb {R}}}W(t)- W_{M,N}(t))\Vert ^2_{{ \mathbb {R}}}+\frac{1}{16} \Vert \partial _v\widetilde{W}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}\\& \le c\big (\Vert W(t)\Vert ^2_{\infty }+ (\beta N+M^2)({\mathscr {B}}^{1,1}_{\beta ,{ \mathbb {R}}}(W(t))^2+\Vert W(t)\Vert ^2_{{ \mathbb {R}}})\big )\\&\quad \times (\Vert v(I_{M-1,N,\alpha ,\beta ,{ \mathbb {R}}}W(t)-W(t))\Vert ^2_{{ \mathbb {R}}} +\Vert v(W(t)-W_{N}(t))\Vert ^2_{{ \mathbb {R}}})+\frac{1}{16} \Vert \partial _v\widetilde{W}_{N}(t)\Vert ^2_{{ \mathbb {R}}}\\& \le c\big ((\beta N)^{{1-r}}+M^{-2q}\big )\big (\Vert W(t)\Vert ^2_{\infty }+ (\beta N+M^2)({\mathscr {B}}^{1,1}_{\beta ,{ \mathbb {R}}}(W(t))^2+\Vert W(t)\Vert ^2_{{ \mathbb {R}}})\big )\\&\quad \times \big ({\mathscr {B}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(t))^2+{\mathbb {C}}^{q,r}_{N,0,\beta ,{ \mathbb {R}}}(W(t))+{\mathbb {C}}^{q,r}_{N,2,\beta , { \mathbb {R}}}(W(t))\big ) +\dfrac{1}{16}||\partial _v\widetilde{W}_{N}(t)||_{{ \mathbb {R}}}^{2}. \end{aligned}$$
Combining of the above two formulations, we get that
$$\begin{aligned} & |G_{4}(t,\widetilde{W}_{N}(t))|\\& \le \dfrac{1}{8}||\partial _v\widetilde{W}_{N}(t)||_{{ \mathbb {R}}}^{2} + c\big ((\beta N)^{{1-r}}+M^{-2q}\big )\Big (\big (\Vert W(t)\Vert ^2_{\infty }+ (\beta N+M^2)({\mathscr {B}}^{1,1}_{\beta ,{ \mathbb {R}}}(W(t))^2+\Vert W(t)\Vert ^2_{{ \mathbb {R}}})\big )\\&\quad \times \big ({\mathscr {B}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(t))^2+{\mathbb {C}}^{q,r}_{N,0, \beta , { \mathbb {R}}}(W(t))+{\mathbb {C}}^{q,r}_{N,2, \beta ,{ \mathbb {R}}}(W(t))\big ) +{\mathbb {C}}^{q,r}_{N,0, \beta ,{ \mathbb {R}}}(W^2(t))+{\mathbb {C}}^{q,r}_{N,2, \beta ,{ \mathbb {R}}}(W^2(t))\Big ). \end{aligned}$$
(47)
By (19) and (20) with \(\alpha =0\), we deduce that
$$\begin{aligned} |G_{5}(t,\widetilde{W}_{M,N}(t))|&=2|k(v (W^2_{M,N}(t)-w^2_{M,N}(t)), \partial _v\widetilde{W}_{M,N}(t))_{M,N,{ \mathbb {R}}}|\\&=2|(v\widetilde{W}^2_{M,N}(t)+2v\widetilde{W}_{M,N}(t)W(t), \partial _v\widetilde{W}_{M,N}(t))_{M,N,{ \mathbb {R}}}\\&\quad +(2v \widetilde{W}_{M,N}(t)( W_{M,N}(t)- I_{M,N,\alpha ,\beta ,{ \mathbb {R}}}W(t)), \partial _v\widetilde{W}_{M,N}(t))_{M, N,{ \mathbb {R}}}|. \end{aligned}$$
Using (33) and (34) with \(q=\infty\), we drive that
$$\begin{aligned} & |(v\widetilde{W}^2_{M,N}(t), \partial _v\widetilde{W}_{M,N}(t))_{M,N,{ \mathbb {R}}}|\\&\quad \le c(\beta ^{-1} N+M)(\sqrt{\beta N}+M)\Vert \widetilde{W}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}||\partial _v\widetilde{W}_{M,N}(t)||_{{ \mathbb {R}}},\\&\quad \le c(\beta ^{-1} N+M)(\sqrt{\beta N}+M) \Vert \widetilde{W}_{M,N}(t)\Vert _{{ \mathbb {R}}}(\Vert \widetilde{W}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}+ ||\partial _v\widetilde{W}_{M,N}(t)||^2_{{ \mathbb {R}}}),\\&|(2v\widetilde{W}_{M,N}(t)W(t), \partial _v\widetilde{W}_{M,N}(t))_{M,N,{ \mathbb {R}}}|\le c\Vert v W(t)\Vert ^2_{\infty }\Vert \widetilde{W}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}} +\dfrac{1}{8} ||\partial _v\widetilde{W}_{M,N}(t)||_{{ \mathbb {R}}}^{2}, \end{aligned}$$
and
$$\begin{aligned} &|(2v \widetilde{W}_{M,N}(t)( W_{M,N}(t)- I_{M,N,\alpha ,\beta ,{ \mathbb {R}}}W(t)), \partial _v\widetilde{W}_{M,N}(t))_{M,N,{ \mathbb {R}}}|\\& \le 2 \Vert \widetilde{W}_{M,N}(t)\Vert _{L^\infty ({ \mathbb {R}})}\Vert v( W_{M,N}(t)- I_{M,N,\alpha ,\beta ,{ \mathbb {R}}}W(t))\Vert _{M,N,{ \mathbb {R}}}\Vert \partial _v\widetilde{W}_{M,N}(t)\Vert _{M,N,{ \mathbb {R}}}\\& \le c\big (\beta N+M^2\big ){\bar{d}}(M,N,0)^2 \Vert \widetilde{W}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}\Vert v( W_{M,N}(t)- I_{M,N,\alpha ,\beta ,{ \mathbb {R}}}W(t))\Vert ^2_{{ \mathbb {R}}}+ \frac{1}{8} \Vert \partial _v\widetilde{W}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}\\& \le \dfrac{1}{8}\Vert \partial _v\widetilde{W}_{N}(t)\Vert ^2_{{ \mathbb {R}}} + c\big ((\beta N)^{{1-r}}+M^{-2q}\big )\big (\beta N+M^2\big ){\bar{d}}(M,N,0)^2 \Vert \widetilde{W}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}\\&\quad \times \big ({\mathscr {B}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(t))^2+{\mathbb {C}}^{q,r}_{N,0, \beta ,{ \mathbb {R}}}(W(t))+{\mathbb {C}}^{q,r}_{N,2,\beta ,{ \mathbb {R}}}(W(t))\big ). \end{aligned}$$
For \(r>1\) and \(q>1\), we have
$$\begin{aligned} |G_{5}(t,\widetilde{W}_{N}(t))|&\le c(\beta ^{-1} N+M)(\sqrt{\beta N}+M) \Vert \widetilde{W}_{M,N}(t)\Vert _{{ \mathbb {R}}}(\Vert \widetilde{W}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}+ ||\partial _v\widetilde{W}_{M,N}(t)||^2_{{ \mathbb {R}}})\\&\quad +c(\Vert v W(t)\Vert _\infty +\tilde{d}(W(t),\beta )) ||\widetilde{W}_{M,N}(t))||^2_{{ \mathbb {R}}}+\dfrac{1}{4} ||\partial _v\widetilde{W}_{M,N}(t)||^2_{{ \mathbb {R}}}, \end{aligned}$$
(48)
where \(\tilde{d}(W(t),\beta )\) is a positive constant depending only on \(||\partial _v^r(\text{e}^{\frac{1}{2} \beta (|v|-1)}u) ||^2_{\bar{\omega }_{i,\beta }, \,\Lambda _1\cup \Lambda _2}\) (\(i=r-1,r,r+1,r+2\)), \(||\partial _v^{r+1}(\text{e}^{{\frac{1}{2}}\beta (|v|-1)} W(t))||_{\bar{\omega }_{r+1,\beta },\Lambda _1\cup \Lambda _2}\), \(||(1-v^2)^{\frac{q-1}{2}}\partial _v^{q}{u}||^2_{I}\) and \(\beta\).
Finally, by (5), (10), (15) and the definition of discrete inner product (18), we get that
$$\begin{aligned} |G_{6}(t,2\widetilde{W}_{N}(t))| \le 18||\widetilde{W}_{M,N}(t)||^2_{{ \mathbb {R}}}+\frac{1}{2} ||\partial _v\widetilde{W}_{M,N}(t)||^2_{{ \mathbb {R}}}. \end{aligned}$$
(49)
By inserting (44)–(49) into (43), we find that
$$\begin{aligned} &\partial _t||\widetilde{W}_{M,N}(t)||^2_{M,N,{ \mathbb {R}}}+||\partial _v \widetilde{W}_{M,N}(t)||_{{ \mathbb {R}}}^2\\ & \le c(\beta ^{-1} N+M)(\sqrt{\beta N}+M) \Vert \widetilde{W}_{M,N}(t)\Vert _{{ \mathbb {R}}}(\Vert \widetilde{W}_{N}(t)\Vert ^2_{{ \mathbb {R}}}+ ||\partial _v\widetilde{W}_{M,N}(t)||^2_{{ \mathbb {R}}})\\&\quad +c\big (\Vert v W(t)\Vert _\infty +\tilde{d}(W(t),\beta )+19\big ) ||\widetilde{W}_{M,N}(t)||^2_{{ \mathbb {R}}} +c\big ((\beta N)^{{1-r}}+M^{2-2q}\big ){ {\mathscr {R}}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(t)), \end{aligned}$$
or equivalently,
$$\begin{aligned} &\partial _t||\widetilde{W}_{M,N}(t)||_{M,N,{ \mathbb {R}}}^2+||\partial _v \widetilde{W}_{M,N}(t)||_{{ \mathbb {R}}}^2+||\widetilde{W}_{M,N}(t)||^2_{{ \mathbb {R}}}\\& \le c(\beta ^{-1} N+M)(\sqrt{\beta N}+M) \Vert \widetilde{W}_{M,N}(t)\Vert _{{ \mathbb {R}}}(\Vert \widetilde{W}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}+ ||\partial _v\widetilde{W}_{M,N}(t)||^2_{{ \mathbb {R}}})\\&\quad +c(\Vert v W(t)\Vert _\infty +\tilde{d}(W(t),\beta )+20) ||\widetilde{W}_{M,N}(t)||^2_{{ \mathbb {R}}} \\&\quad +c\big ((\beta N)^{{1-r}}+M^{2-2q}\big ){ {\mathscr {R}}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(t)), \end{aligned}$$
(50)
where
$$\begin{aligned}&{ {\mathscr {R}}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(t))\\& ={\mathscr {B}}^{q,r}_{\beta ,{ \mathbb {R}}}(\partial _tW(t))^2 +{\mathscr {B}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(t))^2+\Big (\big (\Vert W(t)\Vert ^2_{\infty } +(\beta N+M^2)({\mathscr {B}}^{1,1}_{\beta ,{ \mathbb {R}}}(W(t))^2+\Vert W(t)\Vert ^2_{{ \mathbb {R}}})\big )\\& \quad \times \big ({\mathscr {B}}^{q,r+1}_{\beta ,{ \mathbb {R}}}(W(t))^2 +{\mathbb {C}}^{q,r+1}_{N,0,\beta ,{ \mathbb {R}}}(W(t))+{\mathbb {C}}^{q,r+1}_{N,2,\beta ,{ \mathbb {R}}}(W(t))\big ) +{\mathbb {C}}^{q,r+1}_{N,0, \beta ,{ \mathbb {R}}}(W^2(t))+{\mathbb {C}}^{q,r+1}_{N,2, \beta ,{ \mathbb {R}}}(W^2(t))\Big ). \end{aligned}$$
Integrating (50) with respect to t, we deduce that
$$\begin{aligned} &||\widetilde{W}_{M,N}(t)||_{M,N,{ \mathbb {R}}}^2 +\displaystyle \int ^t_0(1-c\mu (M,N,\beta ) ||\widetilde{W}_{M,N}(\tau ))||_{{ \mathbb {R}}})(||\partial _v \widetilde{W}_{M,N}(\tau )||_{{ \mathbb {R}}}^2+||\widetilde{W}_{M,N}(\tau )||^2_{{ \mathbb {R}}})\text {d}\tau \\& \le c\big ((\beta N)^{{1-r}}+M^{2-2q}\big )\displaystyle \int ^t_0 { {\mathscr {R}}}^{q,r}_{\beta , { \mathbb {R}}}(W(\tau )){\text {d}}\tau +c\lambda \big ((\beta N)^{{1-r}}+M^{2-2q}\big )\big ({\mathscr {B}}^{q,r}_{\beta ,{ \mathbb {R}}}(W_0)^2+{\mathbb {C}}^{q-1,r}_{N,0, \beta ,{ \mathbb {R}}}(W_0)\big )\\&\quad +c \displaystyle \int ^t_0\big (\Vert v W(\tau )\Vert _\infty +\tilde{d}(W(t),\beta )+20\big ) ||\widetilde{W}_{M,N}(\tau )||^2_{{ \mathbb {R}}}\text {d}\tau , \end{aligned}$$
(51)
where \(\mu (M,N,\beta )=(\beta ^{-1} N+M)(\sqrt{\beta N}+M)\), \(\lambda =1\) for \(w_{M,N}(0)=I_{M,N,0,\beta ,{ \mathbb {R}}}W_0\), and \(\lambda =0\) for \(w_{M,N}(0)=P^1_{M,N,\beta , { \mathbb {R}}}W_{0}\).

We shall use the following lemma.

Lemma 3.1

(Lemma 3.1 of [13]) Assume that
  • the constants \(b_1>0, b_2 \ge 0, b_3\ge 0\) and \(d \ge 0,\)

  • Z(t) and A(t) are non-negative functions of t,

  • \(d\le {b^2_1}/{b^2_2}\text{e}^{-b_3t_1}\) for \(t_1>0,\)

  • for all \(t \le t_1,\)
    $$\begin{aligned} Z(t) + \displaystyle \int ^t_0 (b_1-b_2 Z^{\frac{1}{2}}(\tau ))A(\tau )\text{d}\tau \le d+ b_3 \displaystyle \int ^t_0 Z(\tau )\text {d}\tau . \end{aligned}$$
Then, for all\(t\le t_1\),
$$\begin{aligned} Z(t)\le d \text{e}^{b_3t }. \end{aligned}$$
We now take in Lemma 3.1, \(b_1=1, b_2=c\mu (M,N,\beta ), b_3=c(|||v W(t)|||_\infty +\tilde{d}(W(t),\beta )+20),\) with \(|||v W(t)|||_\infty =\underset{0\le t \le T}{\sup }||v W(t)||_\infty\) and
$$\begin{aligned} d= & {} c\big ((\beta N)^{{1-r}}+M^{2-2q}\big )\Big (\lambda \big ({\mathscr {B}}^{q,r}_{\beta ,{ {\mathscr {R}}}}(W_0)^2+{\mathbb {C}}^{q-1,r}_{N,0, \beta ,{ \mathbb {R}}}(W_0)\big )+ \displaystyle \int ^t_0{ {\mathscr {R}}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(\tau ))\text {d}\tau \Big ),\\ Z(t)= & {} ||\widetilde{W}_{M,N}(t)||_{{ \mathbb {R}}}^2+\displaystyle \int ^t_0||\widetilde{W}_{M,N}(\tau ))||^2_{1,{ \mathbb {R}}}\text {d}\tau ,\\ A(t)= & {} ||\partial _v\widetilde{W}_{M,N}(\tau )||_{{ \mathbb {R}}}^2+||\widetilde{W}_{M,N}(\tau ))||^2_{{ \mathbb {R}}}. \end{aligned}$$
Moreover, if \(q,r>1\) and \(||{ {\mathscr {R}}}^{q,r}_{\beta ,{ \mathbb {R}}}(W) ||_{L^1(0,T)}< \infty ,\) then \(d \rightarrow 0\) as \(M,N\rightarrow \infty\). Therefore, applying (21) and Lemma 3.1 to (50), we obtain that for any \(T>0\),
$$\begin{aligned} & ||\widetilde{W}_{M,N}(t)||_{{ \mathbb {R}}}^2+\displaystyle \int ^t_0||\widetilde{W}_{M,N}(\tau ))||^2_{1,{ \mathbb {R}}}\text {d}\tau \\& \le c\text{e}^{b_3T}\big ((\beta N)^{{1-r}}+M^{2-2q}\big )\Big (\lambda \big ({\mathscr {B}}^{q,r}_{\beta ,{ \mathbb {R}}}(W_0)^2+{\mathbb {C}}^{q-1,r}_{N,0, \beta ,{ \mathbb {R}}}(W_0)\big )+ \displaystyle \int ^t_0{ {\mathscr {R}}}^{q,r}_{\beta , { \mathbb {R}}}(W(\tau ))\text {d}\tau \Big ). \end{aligned}$$
(52)
Finally, a combination of (30) and (51) leads to the following result.

Theorem 3.1

For \(0\le t \le T,\) we have
$$\begin{aligned} & ||W(t)-{w}_{M,N}(t)||_{{ \mathbb {R}}}^2+\displaystyle \int ^t_0||W(\tau )-{w}_{M,N}(\tau )||^2_{1,{ \mathbb {R}}}\text {d}\tau \\& \le c\big ((\beta N)^{{1-r}}+M^{2-2q}\big )\Big (\text{e}^{b_3T}\big (\lambda \big ({\mathscr {B}}^{q,r}_{\beta ,{ \mathbb {R}}}(W_0)^2+{\mathbb {C}}^{q-1,r}_{N,0, \beta ,{ \mathbb {R}}}(W_0)\big )+ \displaystyle \int ^t_0{ {\mathscr {R}}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(\tau ))\text {d}\tau \big )\\&\quad +{\mathscr {B}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(t))^2+\displaystyle \int ^t_0{\mathscr {B}}^{q,r}_{\beta ,{ \mathbb {R}}}(W(\tau ))^2\text {d}\tau \Big ), \end{aligned}$$
(53)
provided that the norms appearing in the previous statements are finite.

Remark 3.1

We can use the composite Laguerre spectral method with mode N to solve problem (1), see Wang [37]. Let \(r>1\), for any \(0 \le t \le T\), we have that (cf. Theorem 3.1 of [37])
$$\begin{aligned} &||W(t)-{w}_{N}(t)||_{{ \mathbb {R}}}^2+\displaystyle \int ^t_0||\partial _v(W(\tau )-{w}_{N}(\tau ))(\tau )||_{{ \mathbb {R}}}^2\text {d}\tau \le c_2(\beta + \beta ^{-2})^2(\beta N)^{{1-r}}, \end{aligned}$$
(54)
where \(c_2\) is a positive constant depending on \(c_1,T\) and \(\underset{0\le t \le T}{\max } ||\partial _x^{r+1}(\text{e}^{{\frac{1}{2}}\beta |v|}W(\tau ))||_{\omega _{r+1,\beta },{ \mathbb {R}}},\) etc.. Comparing the result of (54) with (53), the accuracy is improved by the multi-domain decomposition pseudospectral method.

The main idea and techniques used in this proof come from [17, 37] and we generalized them.

3.4 Stability Analysis

We now consider the stability of scheme (37), which might be of the generalized stability as described in [12]. Suppose that \(W_{0}\) has the errors \(\tilde{W}_{0}\). They induce the error of \(w_{M,N}\) denoted by \(\tilde{w}_{M,N}\). Then, we obtain from (37) that for all \(\phi \in \mathcal{Q}_{M,N,\beta }({ \mathbb {R}})\) satisfy
$$\begin{aligned} \left\{ \begin{array}{ll} (\partial _t\tilde{w}_{M,N}(t),\phi )_{M,N,{ \mathbb {R}}}+ (v\tilde{w}_{M,N}(t), \partial _v\phi )_{M,N,{ \mathbb {R}}} +k(v (2{w}_{M,N}(t)\tilde{w}_{M,N}(t)+\tilde{w}^2_{M,N}(t)),\partial _v\phi )_{M,N,{ \mathbb {R}}}\\ +(\partial _v\tilde{w}_{M,N}(t),\partial _v\phi )_{{ \mathbb {R}}} =0,\qquad \qquad \qquad \qquad \forall \phi \in \mathcal{Q}_{M,N,\beta }({ \mathbb {R}}),\quad 0<t\le T,\\ \tilde{w}_{M,N}(0)={\tilde{w}}_{0}. \end{array} \right. \end{aligned}$$
(55)
Taking \(\phi =2\tilde{w}_{M,N}(t)\) in (55), we derive that for \(0 \le t \le T,\)
$$\begin{aligned} & \partial _t||\widetilde{w}_{M,N}(t)||_{M,N,{ \mathbb {R}}}^2+2||\partial _v \widetilde{w}_{M,N}(t)||_{{ \mathbb {R}}}^2\\ & =-2k(v (2{w}_{M,N}(t)\tilde{w}_{M,N}(t)+\tilde{w}^2_{M,N}(t)),\partial _v\tilde{w}_{M,N}(t))_{M,N,{ \mathbb {R}}}\\& \quad -2(v \tilde{w}_{M,N}(t), \partial _v\widetilde{w}_{M,N}(t))_{M,N,{ \mathbb {R}}}. \end{aligned}$$
(56)
Thanks to the Cauchy inequality, (22), (34) with \(q=2\) and Proposition 2.2, it yields that
$$\begin{aligned} & |-2k(v (2{w}_{M,N}(t)\tilde{w}_{M,N}(t)+\tilde{w}^2_{M,N}(t)),\partial _v\tilde{w}_{M,N}(t))_{M,N,{ \mathbb {R}}}|\\&=|2(2v {w}_{M,N}(t)\tilde{w}_{M,N}(t),\partial _v\tilde{w}_{M,N}(t))_{M,N,{ \mathbb {R}}}+(v\tilde{w}^2_{M,N}(t),\partial _v\tilde{w}_{M,N}(t))_{M,N,{ \mathbb {R}}}|\\& \le c \mu (M,N,\beta )||\widetilde{w}_{M,N}(t)||_{{ \mathbb {R}}}(||\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}}+||\partial _v\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}})\\&\quad +2||vw_{M,N}(t)||^2_{L^\infty ({ \mathbb {R}})}||\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}}+\dfrac{1}{2} ||\partial _v\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}}. \end{aligned}$$
(57)
Like (49),
$$\begin{aligned} |-2(v \tilde{w}_{M,N}(t), \partial _v\widetilde{w}_{M,N}(t))_{M,N,{ \mathbb {R}}}|\le 18||\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}}+\dfrac{1}{2} ||\partial _v\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}}. \end{aligned}$$
(58)
Substituting (57)–(58) into (56), we deduce that
$$\begin{aligned} & \partial _t||\widetilde{w}_{M,N}(t)||_{{ \mathbb {R}}}^2+||\partial _v \widetilde{w}_{M,N}(t)||_{{ \mathbb {R}}}^2+||\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}}\\ & \le c\mu (M,N,\beta )||\widetilde{w}_{M,N}(t)||_{{ \mathbb {R}}}(||\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}}+||\partial _v\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}})\\ & \quad +( ||v w_{M,N}(t)||^2_{L^\infty ({ \mathbb {R}})}+19)||\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}}. \end{aligned}$$
(59)
Integrating the inequality (59) from 0 to t with respect to t, we obtain that
$$\begin{aligned} &\Vert \tilde{w}_{M,N}(t)\Vert ^{2}_{{ \mathbb {R}}} +\displaystyle \int ^{t}_{0}\big (1-c\mu (M,N,\beta )||\widetilde{w}_{M,N}(t)||_{{ \mathbb {R}}}\big )(||\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}}+||\partial _v\widetilde{w}_{M,N}(t)||^2_{{ \mathbb {R}}})\text {d}\xi \\& \le \Vert \tilde{w}_{0}\Vert ^{2}_{{ \mathbb {R}}}+\displaystyle \int ^{t}_{0}(||v w_{M,N}(t)||^2_{L^\infty ({ \mathbb {R}})}+19)||\widetilde{w}_{M,N}(\xi )||^2_{{ \mathbb {R}}}\text {d}\xi . \end{aligned}$$
Let Z(t) be the same as in Lemma 3.1. In Lemma 3.1, we take
$$\begin{aligned} d=\Vert \tilde{w}_{0}\Vert ^{2}_{{ \mathbb {R}}},\quad b_1=1,\, b_2=c \mu (M,N,\beta )\,\mathrm{and }\, b_3= |||v w_{M,N}(t)|||^2_\infty+19. \end{aligned}$$
Finally, applying Lemma 3.1, we get the following result of stability.

Theorem 3.2

Suppose that
$$\begin{aligned} \Vert \tilde{w}_{0}\Vert ^{2}_{{ \mathbb {R}}}\le \dfrac{1}{c \mu (M,N,\beta )^2}\displaystyle \exp (-(|||v w_{M,N}(t)|||^2_\infty+19)T), \end{aligned}$$
then for all\(0\le t\le T\)
$$\begin{aligned} \Vert \tilde{w}_{M,N}(t)\Vert ^2_{{ \mathbb {R}}}+\int _0^t\Vert \tilde{w}_{M,N}(\xi )\Vert ^2_{1,{ \mathbb {R}}}\text{d}\xi \le \Vert \tilde{w}_{0}\Vert ^{2}_{{ \mathbb {R}}}\displaystyle \exp (C_{1}(w_{M,N}(t))T), \end{aligned}$$
(60)
where \(C_{1}(w_{M,N}(t))\) depends on \(|||v w_{M, N}(t)|||^2_\infty\).

4 Concluding Remarks

In this paper, we developed a multi-domain decomposition generalized Laguerre–Legendre pseudospectral method for the nonlinear Fokker–Planck equation defined on the whole line. The numerical results demonstrated the spectral accuracy in space and well confirmed the theoretical analysis.

The main advantages of the proposed approach are as follows:
  • Using generalized Laguerre interpolation approximations on unbounded subdomains \(\Lambda _j\,( j=1,2)\), we could deal with non-standard types of PDEs on the whole line properly. This trick also simplifies actual computation, especially for larger modes N.

  • Using the Legendre interpolation approximations on the bounded subdomain I, we could decrease the degrees of the basis used in expansions of the numerical solutions, which benefits the quick convergence of the Legendre expansion in I, while keeps the spectral accuracy.

  • The adjustable parameter \(\beta\) involved in the generalized Laguerre interpolation approximations provides flexibility to match the asymptotic behaviors of the exact solutions as \(|v|\rightarrow \infty\).

We can apply the main idea and techniques developed in this paper to many other nonlinear problems of multiple dimensions. In particular, the results on interpolation approximations are very applicable to various pseudospectral methods with multi-domain decomposition for problems defined on unbounded domains or exterior problems. For instance, let \(I=\{x\,|\,0< x <2\uppi \}\) and \(\Omega =I\times \mathbb {R}\), we may consider the following initial-boundary value problem of the Fokker–Planck equation in an infinite channel (cf. [3])
$$\begin{aligned} \left\{ \begin{array}{ll} \partial _tW(x,v,t)+v\partial _xW(x,v,t)-\beta _0 \partial _v(vW(x,v,t))\\ -\partial _v[(f'(x)-F)W(x,v,t)] -\beta _0 \mu \partial _v^2 W(x,v,t)=0, \qquad &{}(x,v)\in \Omega ,\,\,0<t\le T,\\ W(x+2 \uppi ,v,t)= W(x,v,t), &{}x \in {\bar{I}}, v \in \mathbb {R},\,\,0<t\le T,\\ W(x,v,t)\rightarrow 0, &{} |v|\rightarrow \infty ,\,\,0<t\le T,\\ W(x,v,0)=W_0(x,v),&{}(x,v)\in \bar{\Omega }. \end{array}\right. \end{aligned}$$
(61)
This is a problem with periodic boundary conditions in the x-direction, where \(f(x)=f(x+2\uppi )\) is the periodic potential with a period of \(2\uppi\). Similar to the treatment in this paper, a Legendre approximation can be used on a finite interval, together with Laguerre methods on two sides in the v-direction. We will report our results in the future.

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions and comments, which greatly improve the wording of the paper and also would like to thank Dr. Lu-yu Wang of Zhejiang University for polishing the writing of the paper.

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Copyright information

© Shanghai University 2019

Authors and Affiliations

  1. 1.School of Statistics and MathematicsShanghai Lixin University of Accounting and FinanceShanghaiChina
  2. 2.School of Mathematics and StatisticsHenan University of Science and TechnologyLuoyangChina

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