Structure Preserving Schemes for Nonlinear Fokker–Planck Equations and Applications

Abstract

In this paper we focus on the construction of numerical schemes for nonlinear Fokker–Planck equations that preserve the structural properties, like non negativity of the solution, entropy dissipation and large time behavior. The methods here developed are second order accurate, they do not require any restriction on the mesh size and are capable to capture the asymptotic steady states with arbitrary accuracy. These properties are essential for a correct description of the underlying physical problem. Applications of the schemes to several nonlinear Fokker–Planck equations with nonlocal terms describing emerging collective behavior in socio-economic and life sciences are presented.

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Acknowledgements

The research that led to the present paper was partially supported by the research grant Numerical methods for uncertainty quantification in hyperbolic and kinetic equations of the group GNCS of INdAM. MZ acknowledges “Compagnia di San Paolo”.

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Correspondence to Mattia Zanella.

Appendices

Appendix A: High Order Semi-Implicit Methods

Here we follow the approach in [7]. We write the semi-discrete scheme (7) in the equivalent form

$$\begin{aligned} \begin{aligned} \frac{df_i}{dt}(w,t)&= {{\mathscr {Q}}}_i(\mathbf{f},\mathbf{g}),\\ \frac{dg_i}{dt}(w,t)&= {{\mathscr {Q}}}_i(\mathbf{f},\mathbf{g}),\\ \end{aligned} \end{aligned}$$
(93)

where \(\mathbf{f}=(f_0,\ldots ,f_N), \mathbf{g}=(g_0,\ldots ,g_N)\),

$$\begin{aligned} {{\mathscr {Q}}}_i(\mathbf{f},\mathbf{g}) = \frac{{\mathscr {F}}_{i+1/2}[\mathbf{f},\mathbf{g}]-{\mathscr {F}}_{i-1/2}[\mathbf{f},\mathbf{g}]}{\varDelta w} \end{aligned}$$
(94)

and

$$\begin{aligned} {\mathscr {F}}_{i+1/2}[\mathbf{f},\mathbf{g}]= \tilde{\mathscr {C}}_{i+1/2}[\mathbf{g}] \left[ (1-\delta _{i+1/2}[\mathbf{g}])f_{i+1}+\delta _{i+1/2}[\mathbf{g}] f_i \right] +D_{i+1/2}\dfrac{f_{i+1}-f_i}{\varDelta w}, \end{aligned}$$
(95)

with initial conditions \(f_i(0)=f_0(w_i), g_i(0)=f_0(w_i)\). In the above equations we used the notation \([\cdot ]\) to denote the functional dependence.

System (93) is then solved by an implicit-explicit (IMEX) Runge-Kutta method [28] where the variables \({f}_i\) are treated implicitly and the variables \({g}_i\) are treated explicitly. More precisely, using standard notations we can write an implicit-explicit Runge-Kutta scheme for (93) as follows. First we set \(f_i^n=g_i^n\) and compute for \(h=1,\ldots ,s\)

$$\begin{aligned} \left\{ \begin{aligned} F_i^h&=f_i^n+\varDelta t \sum _{k=1}^{h} a_{hk} {{\mathscr {Q}}}_i(\mathbf{F}^k,\mathbf{G}^k) ,\\ G_i^h&=f_i^n+\varDelta t \sum _{k=1}^{h-1} {{\tilde{a}}}_{hk} {{\mathscr {Q}}}_i(\mathbf{F}^k,\mathbf{G}^k), \end{aligned} \right. \end{aligned}$$
(96)

where \(\mathbf{F}^k = (F_0^k,\ldots ,F_N^k), \mathbf{G}^k = (G_0^k,\ldots ,G_N^k)\) and next we update the numerical solution

$$\begin{aligned} \left\{ \begin{aligned} f_i^{n+1}&=f_i^n+\varDelta t \sum _{k=1}^{s} b_k {{\mathscr {Q}}}_i(\mathbf{F}^k,\mathbf{G}^k) ,\\ g_i^{n+1}&=f_i^n+\varDelta t \sum _{k=1}^{s} {{\tilde{b}}}_{k} {{\mathscr {Q}}}_i(\mathbf{F}^k,\mathbf{G}^k). \end{aligned} \right. \end{aligned}$$
(97)

In particular, the IMEX scheme is chosen such that \(b_k={{\tilde{b}}}_k, k=1,\ldots ,s\) so that \(f^{n+1}=g^{n+1}\) and therefore the duplication of the system is only apparent since there is only one set of numerical solutions. In our numerical tests we coupled the structure preserving discretizations with the second order semi-implicit scheme obtained as a combination of Heun method (explicit) and Crank-Nicolson (implicit) characterized by \(s=2\) and

$$\begin{aligned} a_{11}=0,\quad a_{21}=a_{22}=1/2,\quad {\tilde{a}}_{21}=1,\quad b_k={{\tilde{b}}}_k=1/2,\quad k=1,2. \end{aligned}$$
(98)

As observed in [7] this represents a natural choice when dealing with convection-diffusion type equations, since the Heun method is an SSP explicit RK method, and Crank-Nicolson is an A-stable method, widely used for diffusion problems. Higher order methods can be found in [7, 28].

Appendix B: The Multi-Dimensional Case

In this section we report for the sake of completeness the details of the numerical schemes for multi-dimensional situations. We consider the case of Chang–Cooper type fluxes, and to keep notations simple we restrict to two dimensional problems \(d=2\). We introduce a uniform mesh \((w_i,v_j) \in \varOmega \subseteq \mathbb {R}^2\), with \(\varDelta w=w_{i+1}-w_{i}\) and \(\varDelta v=v_{j+1}-v_{j}\). We denote by \(w_{i+1/2}=w_i+\varDelta w/2\) and \(v_{j+1/2}=v_j+\varDelta v/2\). Let \(f_{i,j}(t)\) be an approximation of the solution \(f(w_i,v_j,t)\) and consider the following discretization of the nonlinear Fokker–Planck equation (4)

$$\begin{aligned} \frac{d}{dt} f_{i,j} = \dfrac{\mathscr {F}_{i+1/2,j}[f]-\mathscr {F}_{i-1/2,j}[f]}{\varDelta w}+\dfrac{\mathscr {F}_{i,j+1/2}[f]-\mathscr {F}_{i,j-1/2}[f]}{\varDelta v}, \end{aligned}$$
(99)

being \(\mathscr {F}_{i\pm 1/2,j}[f], \mathscr {F}_{i,j\pm 1/2}[f]\) flux functions characterizing the numerical discretization. The quasi-stationary approximations over the cell \([w_i,w_{i+1}]\times [v_i,v_{i+1}]\) of the two dimensional problem now read

$$\begin{aligned} \begin{aligned} \int _{w_{i}}^{w_{i+1}} \dfrac{1}{f(w,v_j,t)}\partial _w f(w,v_j,t) dw&= -\int _{w_{i}}^{w_{i+1}}\dfrac{\mathscr {B}[f](w,v_j,t)+\partial _w D(w,v_j)}{D(w,v_j)}dw, \\ \int _{v_{j}}^{v_{j+1}} \dfrac{1}{f(w_i,v,t)}\partial _v f(w_i,v,t) dv&= -\int _{v_j}^{v_{j+1}}\dfrac{\mathscr {B}[f](w_i,v,t)+\partial _v D(w_i,v)}{D(w_i,v)}dv. \end{aligned} \end{aligned}$$

Therefore, setting

$$\begin{aligned} \begin{aligned} \tilde{\mathscr {C}}_{i+1/2,j}&= \dfrac{D_{i+1/2,j}}{\varDelta w}\int _{w_i}^{w_{i+1}}\dfrac{\mathscr {B}[f](w,v_j,t)+\partial _{w}D(w,v_j)}{D(w,v_j)}dw\\ \tilde{\mathscr {C}}_{i,j+1/2}&= \dfrac{D_{i,j+1/2}}{\varDelta v}\int _{v_j}^{v_{j+1}}\dfrac{\mathscr {B}[f](w_i,v,t)+\partial _{v}D(w_i,v)}{D(w_i,v)}dv \end{aligned} \end{aligned}$$
(100)

and by considering the natural generalization of the one-dimensional numerical flux

$$\begin{aligned} \begin{aligned} \mathscr {F}_{i+1/2,j}[f]&= \tilde{\mathscr {C}}_{i+1/2,j}\tilde{f}_{i+1/2,j}+D_{i+1/2,j}\dfrac{f_{i+1,j}-f_{i,j}}{\varDelta w}\\ {\tilde{f}}_{i+1/2,j}&= (1-\delta _{i+1/2,j})f_{i+1,j}+\delta _{i+1/2,j}f_{i,j}\\ \mathscr {F}_{i,j+1/2}[f]&= \tilde{\mathscr {C}}_{i,j+1/2}\tilde{f}_{i,j+1/2}+D_{i,j+1/2}\dfrac{f_{i,j+1}-f_{i,j}}{\varDelta v}\\ {\tilde{f}}_{i,j+1/2}&= (1-\delta _{i,j+1/2})f_{i,j+1}+\delta _{i,j+1/2}f_{i,j}, \end{aligned} \end{aligned}$$
(101)

we define \(\delta _{i+1/2,j}\) and \(\delta _{i,j+1/2}\) in such a way that we preserve the steady state solution for each dimension. The Chang–Cooper type structure preserving methods are then given by

$$\begin{aligned} \begin{aligned} \delta _{i+1/2,j}&= \dfrac{1}{\lambda _{i+1/2,j}}+\dfrac{1}{1-\exp (\lambda _{i+1/2,j})},\\ \lambda _{i+1/2,j}&= \dfrac{\varDelta w \tilde{\mathscr {C}}_{i+1/2,j}}{D_{i+1/2,j}} \end{aligned} \end{aligned}$$
(102)

and

$$\begin{aligned} \begin{aligned} \delta _{i,j+1/2}&= \dfrac{1}{\lambda _{i,j+1/2}}+\dfrac{1}{1-\exp (\lambda _{i,j+1/2})},\\ \lambda _{i,j+1/2}&= \dfrac{\varDelta v \tilde{\mathscr {C}}_{i,j+1/2}}{D_{i,j+1/2}}. \end{aligned} \end{aligned}$$
(103)

The cases of higher dimension \(d\ge 3\) and entropic average fluxes may be derived in a similar way.

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Pareschi, L., Zanella, M. Structure Preserving Schemes for Nonlinear Fokker–Planck Equations and Applications. J Sci Comput 74, 1575–1600 (2018). https://doi.org/10.1007/s10915-017-0510-z

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Keywords

  • Structure preserving methods
  • Finite difference schemes
  • Fokker–Planck equations
  • Emerging collective behavior