Fast conservative numerical algorithm for the coupled fractional Klein-Gordon-Schrödinger equation

Abstract

In this paper, we are concerned with the numerical solutions of the coupled fractional Klein-Gordon-Schrödinger equation. The numerical schemes are constructed by combining the Crank-Nicolson/leap-frog difference methods for the temporal discretization and the Galerkin finite element methods for the spatial discretization. We give a detailed analysis of the conservation properties in the senses of discrete mass and energy. Then the numerical solutions are shown to be unconditionally bounded in L² −norm, $H^{\frac {\alpha }{2}}-$semi-norm and $L^{\infty }-$norm, respectively. Based on the well-known Brouwer fixed-point theorem and the mathematical induction, the unique solvability of the discrete solutions is proved. Moreover, the schemes are proved to be unconditionally convergent with the optimal order $O\left (\tau ^{2}+h^{r+1}\right )$, where τ is the temporal step, h is the spatial grid size, and r is the order of the selected finite element space. Furthermore, by using the proposed decoupling and iterative algorithms, several numerical examples are included to support theoretical results and show the effectiveness of the schemes. Finally, the fast Krylov subspace solver with suitable circulant preconditioner is designed to effectively solve the Toeplitz-like linear systems. In each iterative step, this method can effectively reduce the memory requirement of above each finite element scheme from ${{O}\left (M^{2}\right )}$ to O(M), and the computational complexity from ${O\left (M^{3}\right )}$ to ${O(M \log M)}$, where M is the number of grid nodes. Numerical tests are carried out to show that this fast algorithm is more practical than the traditional backslash and LU factorization/Cholesky decomposition methods, in terms of memory requirement and computational cost.

Appendix: Proof of Theorem 4.4

Let us prove this theorem step by step.

• (I) For the unique existence of Φ¹, let us consider the homogeneous system of (3.7), i.e.,

  $$ \frac{2}{\tau^{2}}({\Phi}^{1}, \omega_{h}) + \gamma \mathcal{B}({\Phi}^{1}, \omega_{h}) +\eta^{2}({\Phi}^{1}, \omega_{h}) = 0, \quad \forall\omega_{h}\in {S_{h}^{r}}. $$

  (A.1)

  Substituting $\omega _{h}={\Phi }^{1}$ in (A.1) results in the following:

  $$ \left( 1+\frac{\eta^{2}\tau^{2}}{2}\right)\|{\Phi}^{1}\|^{2}+\gamma|{\Phi}^{1}|^{2}_{\frac{\alpha}{2}}=0. $$

  (A.2)

  Hence, one obtains Φ¹ = 0, which shows (3.7) is uniquely solvable.

• (II) To prove the existence of the discrete system (3.5), the Brouwder fixed-point theorem is adopted. Rewrite (3.5) into its equivalent form as follows:

  $$ \begin{array}{@{}rcl@{}} i(\widehat{U}^{n+\frac{1}{2}}, v_{h})-\frac{\lambda\tau}{4}\mathcal{B}(\widehat{U}^{n+\frac{1}{2}}, v_{h}) +\frac{\tau}{2}\left( \left[\kappa_{1}+\kappa_{2}\left( |2\widehat{U}^{n+\frac{1}{2}}-U^{n}|^{2}+|U^{n}|^{2}\right)\right]\right.\\ \left. \times \widehat{U}^{n+\frac{1}{2}}\widehat{\Phi}^{n+\frac{1}{2}} , v_{h}\right)-i(U^{n},v_{h})=0. \end{array} $$

  (A.3)

  Denote $\varPsi : {S_{h}^{r}}\rightarrow {S_{h}^{r}}$ as follows:

  $$ \begin{array}{@{}rcl@{}} &&({\varPsi}(\varpi), v_{h}) := \frac{\lambda}{4}\mathcal{B}(\varpi, v_{h})-\frac{1}{2}\left( \left[\kappa_{1}+\kappa_{2}\left( |2\varpi-U^{n}|^{2}+|U^{n}|^{2}\right)\right]\varpi\widehat{\Phi}^{n+\frac{1}{2}} , v_{h}\right),\\ &&\varpi\in {S_{h}^{r}}. \end{array} $$

  (A.4)

  It is obvious that in order to prove the existence of (3.5), we only need to prove the existence of the following one as follows:

  $$ \widehat{U}^{n+\frac{1}{2}} = U^{n}-i\tau {\varPsi}\left( \widehat{U}^{n+\frac{1}{2}}\right). $$

  (A.5)

  To this end, let us denote $\mathcal {F}: {S_{h}^{r}}\rightarrow {S_{h}^{r}}$ as follows:

  $$ \mathcal{F}(\varpi):= \varpi-U^{n}+i\tau {\varPsi}(\varpi). $$

  (A.6)

From Lemma 4.4, one intends to conclude $Re(\mathcal {F}(\varpi ), \varpi )\geq 0$. Indeed, as the result of the following:

$$Im({\varPsi}(\varpi), \varpi) = 0, $$

we have the following:

$$ \begin{array}{@{}rcl@{}} Re(\mathcal{F}(\varpi), \varpi) &=& \|\varpi\|^{2}-Re(U^{n}, \varpi) \\ &\geq& \|\varpi\|^{2} - \|U^{n}\|\cdot \|\varpi\| \\ &=& \|\varpi\|(\|\varpi\|-\|U^{n}\|). \end{array} $$

(A.7)

If we set ∥ϖ∥ = ∥Uⁿ∥ in (A.7), which is a constant by Theorem 4.1, then it follows from (A.7):

$$ Re(\mathcal{F}(\varpi), \varpi)\geq 0. $$

(A.8)

Accordingly, we complete the proof of the existence of the solution to (3.5).

Next, we prove the uniqueness of the solution Uⁿ, 0 ≤ n ≤ N. Assume there are two solutions $X^{n+1}, Y^{n+1}\in {S_{h}^{r}}$ to solve the discrete scheme (3.5). Then, by (A.5), one has the following:

$$ \|\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\|^{2} = -i\tau \left( {\varPsi}\left( \widehat{X}^{n+\frac{1}{2}}\right) -{\varPsi}\left( \widehat{Y}^{n+\frac{1}{2}}\right), \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\right). $$

(A.9)

From (3.5)–(3.7), we notice that when one intends to find the solution U^n+ 1, the solutions Uⁿ and Φ^n+ 1 have been solved. Then, taking the real part of (A.9), we obtain the following:

$$ \begin{array}{@{}rcl@{}} &&\|\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\|^{2}\\&=& \tau \text{Im}\left\{\frac{1}{2}\left( \left[\kappa_{1}+\kappa_{2}\left( |2\widehat{X}^{n+\frac{1}{2}}-U^{n}|^{2}+|U^{n}|^{2}\right)\right]\widehat{X}^{n+\frac{1}{2}}\widehat{\Phi}^{n+\frac{1}{2}} , \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\right)\right\} \\ && - \tau \text{Im}\left\{\frac{1}{2}\left( \left[\kappa_{1}+\kappa_{2}\left( |2\widehat{Y}^{n+\frac{1}{2}} - U^{n}|^{2}+|U^{n}|^{2}\right)\right]\widehat{Y}^{n+\frac{1}{2}}\widehat{\Phi}^{n+\frac{1}{2}} , \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\right)\right\} \\ &=&\frac{\kappa_{1}\tau}{2}\text{Im}\left( (\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}})\widehat{\Phi}^{n+\frac{1}{2}}, \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\right)\\&&+\frac{\kappa_{2}\tau}{2}\text{Im}\left( \mathcal{G}^{n}\widehat{\Phi}^{n+\frac{1}{2}}, \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\right), \\ && \end{array} $$

(A.10)

where

$$ \mathcal{G}^{n}:=\left( |2\widehat{X}^{n+\frac{1}{2}}-U^{n}|^{2}+ |U^{n}|^{2}\right)\widehat{X}^{n+\frac{1}{2}}-\left( |2\widehat{Y}^{n+\frac{1}{2}}-U^{n}|^{2}+|U^{n}|^{2}\right)\widehat{Y}^{n+\frac{1}{2}}. $$

From (4.16), the first term on the RHS of (A.10) can be bounded as follows:

$$ \left| \frac{\kappa_{1}\tau}{2}\text{Im}\left( \left( \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\right)\widehat{\Phi}^{n+\frac{1}{2}}, \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\right)\right|\leq \frac{\kappa_{1}C_{M_{4}}\tau}{2}\|\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}})\|^{2}. $$

(A.11)

Meanwhile, by Lemma 4.5, the second term on the RHS of (A.10) can be bounded as follows:

$$ \begin{array}{@{}rcl@{}} && \left|\frac{\kappa_{2}\tau}{2}\text{Im}\left( \mathcal{G}^{n}\widehat{\Phi}^{n+\frac{1}{2}}, \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\right)\right| \\ & \leq& \left|\frac{\kappa_{2}\tau}{2}\text{Im}\left( |U^{n}|^{2}(\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}})\widehat{\Phi}^{n+\frac{1}{2}}, \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\right) \right|\\&&+\left|\frac{\kappa_{2}\tau}{2}\text{Im}\left( \left[|2\widehat{X}^{n+\frac{1}{2}}-U^{n}|^{2}\widehat{X}^{n+\frac{1}{2}} -|2\widehat{Y}^{n+\frac{1}{2}}-U^{n}|^{2}\widehat{Y}^{n+\frac{1}{2}}\right]\widehat{\Phi}^{n+\frac{1}{2}}, \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\right) \right| \\ &\leq& \frac{\kappa_{2}C_{M_{3}}^{2}C_{M_{4}}\tau}{2}\|\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\|^{2} \\&&+\frac{5\kappa_{2}C_{M_{4}}\tau}{2}\left( \max\left\{|2\widehat{X}^{n+\frac{1}{2}}-U^{n}|, |2\widehat{Y}^{n+\frac{1}{2}}-U^{n}|, |\widehat{X}^{n+\frac{1}{2}}|, |\widehat{Y}^{n+\frac{1}{2}}|\right\}\right)^{2}\|\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\|^{2}\\ &\leq& 23\kappa_{2}C_{M_{3}}^{2}C_{M_{4}}\tau\|\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\|^{2}. \end{array} $$

(A.12)

Then, substituting (A.11) and (A.12) into (A.10) gives the following:

$$ \begin{array}{@{}rcl@{}} \|\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\|^{2}&\leq& \left( \frac{\kappa_{1}C_{M_{4}}}{2}+ 23\kappa_{2}C_{M_{3}}^{2}C_{M_{4}}\right)\tau \|\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\|^{2}\\&:=&C_{1}\tau\|\widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}\|^{2}. \end{array} $$

(A.13)

Whenever τ is small enough such that 0 < C₁τ < 1, we have the following:

$$ \widehat{X}^{n+\frac{1}{2}}-\widehat{Y}^{n+\frac{1}{2}}=0, $$

which implies the following:

$$ X^{n+1}-Y^{n+1}=-(X^{n}-Y^{n})=\cdots=(-1)^{n+1}(X^{0}-Y^{0})=0. $$

(A.14)

From (A.14) and the mathematical induction, we complete the proof of the uniqueness of the solution to the discrete scheme (3.5).

• (III) To prove the unique solvability of the scheme (3.6), we intend to consider the following homogeneous system:

  $$ \left( 1+\frac{\eta^{2}\tau^{2}}{2}\right)({\Phi}^{n+1}, \omega_{h}) + \frac{\gamma\tau^{2}}{2}\mathcal{C}({\Phi}^{n+1}, \omega_{h})=0, \quad \forall\omega_{h}\in {S_{h}^{r}}. $$

  (A.15)

  Denoting $\omega _{h}={\Phi }^{n+1}$ in (A.15) reads the following:

  $$ \left( 1+\frac{\eta^{2}\tau^{2}}{2}\right)\|{\Phi}^{n+1}\|^{2} + \frac{\gamma\tau^{2}}{2}|{\Phi}^{n+1}|_{\frac{\beta}{2}}^{2}=0, $$

  (A.16)

  which further implies that Φ^n+ 1 = 0. Hence, the scheme (3.6) is uniquely solvable.

From (I)–(III), we complete the proof of Theorem 4.4.