A High-Order Algorithm for Time-Caputo-Tempered Partial Differential Equation with Riesz Derivatives in Two Spatial Dimensions

Abstract

A novel second-order numerical approximation for the Riemann–Liouville tempered fractional derivative, called the tempered fractional-compact difference formula is derived by using the tempered Grünwald difference operator and its asymptotic expansion. Using the relationship between Riemann–Liouville and the Caputo tempered fractional derivatives, then the constructed approximation formula is applied to deal with the time-Caputo-tempered partial differential equation in time, while the spatial Riesz derivative are discretized by the fourth-order compact numerical differential formulas. By using the energy method, it is proved that the proposed algorithm to be unconditionally stable and convergent with order ${\mathcal {O}}\left( \tau ^2+h_1^4+h_2^4\right)$, where $\tau$ is the temporal stepsize and $h_1,h_2$ are the spatial stepsizes respectively. Finally, some numerical examples are performed to testify the effectiveness of the obtained algorithm.

Appendix

Below, we consider the Eq. (1) with nonzero initial value condition,

$$\begin{aligned} \displaystyle u(x,y,0)=\psi (x,y),\quad (x,y)\in {\overline{\varOmega }}=[a,b]\times [c,d]. \end{aligned}$$

(35)

Based on Eq. (4), then Eq. (1) can be further transformed into

$$\begin{aligned} \displaystyle \,_{RL}{D}_{0,t}^{\alpha ,\lambda }u(x,y,t)=\kappa _\beta \frac{\partial ^\beta u(x,y,t)}{\partial {|x|^\beta }}+\kappa _\gamma \frac{\partial ^\gamma u(x,y,t)}{\partial {|y|^\gamma }}+\frac{t^{-\alpha }e^{-\lambda t}}{\varGamma (1-\alpha )}\psi (x,y)+f(x,y,t).\nonumber \\ \end{aligned}$$

(36)

Using the previous method, we can obtain the following difference scheme with convergence order ${\mathcal {O}}\left( \tau ^2+h_1^4+h_2^4\right)$ for Eq. (36) with (35) and (3),

$$\begin{aligned} \displaystyle {\mathscr {H}}_x{\mathscr {H}}_y{\mathcal {B}}_{1}^{\alpha ,\lambda }u_{i,j}^k= & {} \kappa _\beta {\mathscr {L}}{\mathscr {H}}_y\delta _x^\beta u_{i,j}^k+\kappa _\gamma {\mathscr {L}} {\mathscr {H}}_x\delta _y^\gamma u_{i,j}^k \nonumber \\&+\,{\mathscr {L}}{\mathscr {H}}_x{\mathscr {H}}_yf_{i,j}^k+{\mathscr {L}}{\mathscr {H}}_x{\mathscr {H}}_yg_{i,j}^k , \nonumber \\&(x_i,y_j)\in \varOmega _h,\quad 1\le k\le N, \end{aligned}$$

(37)

$$\begin{aligned} \displaystyle u_{i,j}^0= & {} \psi (x_i,y_j),\quad (x_i,y_j)\in {\overline{\varOmega }}_h, \end{aligned}$$

(38)

$$\begin{aligned} \displaystyle u_{i,j}^k= & {} \phi (x_i,y_j,t_k),\quad (x_i,y_j)\in \partial \varOmega _h,\quad 1\le k\le N, \end{aligned}$$

(39)

and their corresponding matrix form can be constructed as follows,

$$\begin{aligned}&\displaystyle \left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta }\right) \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathcal {B}}_{1}^{\alpha ,\lambda }\mathbf {{U}}^{k}\right) =\kappa _\beta \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {D}}}_{\beta }\right) \left( {\mathscr {L}}\mathbf {{U}}^{k}\right) \nonumber \\&\quad +\,\kappa _\gamma \left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta }\right) \left( {{\mathbf {D}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathscr {L}}\mathbf {{U}}^{k}\right) +\left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta }\right) \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathscr {L}}\mathbf {{F}}^{k}\right) \nonumber \\&\quad +\,\left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta }\right) \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathscr {L}}\mathbf {{G}}^{k}\right) ,\quad 1\le k\le N, \end{aligned}$$

(40)

$$\begin{aligned}&\displaystyle \mathbf {{U}}^0={\varvec{\Psi }},\quad (x_i,y_j)\in {{\overline{\varOmega }}}_{h}, \end{aligned}$$

(41)

$$\begin{aligned}&\displaystyle \mathbf {{U}}^k={\varvec{\Phi }},\quad (x_i,y_j)\in \partial {\varOmega }_{h},\quad 0\le k\le N, \end{aligned}$$

(42)

where

$$\begin{aligned} g_{i,j}^k= & {} \frac{{t_k}^{-\alpha }e^{-\lambda {t_k}}}{\varGamma (1-\alpha )}\psi (x_i,y_j), \\ \mathbf {{G}}= & {} (g_{1,1},g_{2,1},\ldots ,g_{M_1-1,1},\ldots , g_{1,M_2-1},g_{2,M_2-1},\ldots ,g_{M_1-1,M_2-1})^{T}, \\ {\varvec{\Psi }}= & {} (\psi _{1,1},\psi _{2,1},\ldots ,\psi _{M_1-1,1},\ldots , \psi _{1,M_2-1},\psi _{2,M_2-1},\ldots ,\psi _{M_1-1,M_2-1})^{T}, \end{aligned}$$

and

$$\begin{aligned} {\varvec{\Phi }}=(\phi _{1,1},\phi _{2,1},\ldots ,\phi _{M_1-1,1}, \ldots ,\phi _{1,M_2-1},\phi _{2,M_2-1},\ldots ,\phi _{M_1-1,M_2-1})^{T}. \end{aligned}$$

For the difference scheme (40) with (41) and (42), we can get similar stability result, hat is to say, the difference scheme (40)–(42) is unconditionally stable with respect to the initial value and the source term.

Next, we give a simple but tedious proof. Taking the inner product of Eq. (40) with ${\mathscr {L}}\mathbf {{U}}^{k}$ leads to

$$\begin{aligned}&\displaystyle h_1h_2\left( {\mathscr {L}}\mathbf {{U}}^{k}\right) ^T \left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta }\right) \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathcal {B}}_{1}^{\alpha ,\lambda }\mathbf {{U}}^{k}\right) \nonumber \\&\quad =\kappa _\beta h_1h_2\left( {\mathscr {L}}\mathbf {{U}}^{k}\right) ^T \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {D}}}_{\beta }\right) \left( {\mathscr {L}}\mathbf {{U}}^{k}\right) \nonumber \\&\qquad +\,\kappa _\gamma h_1h_2\left( {\mathscr {L}} \mathbf {{U}}^{k}\right) ^T\left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta }\right) \left( {{\mathbf {D}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathscr {L}}\mathbf {{U}}^{k}\right) \nonumber \\&\qquad +\,h_1h_2\left( {\mathscr {L}}\mathbf {{U}}^{k}\right) ^T\left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta } \right) \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathscr {L}}\mathbf {{F}}^{k}\right) \nonumber \\&\qquad +\,h_1h_2\left( {\mathscr {L}}\mathbf {{U}}^{k}\right) ^T\left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta } \right) \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathscr {L}}\mathbf {{G}}^{k}\right) ,\quad 1\le k\le N. \end{aligned}$$

(43)

Like inequality (27), we also can get

$$\begin{aligned} \displaystyle h_1h_2\left( {\mathscr {L}}\mathbf {{U}}^{k}\right) ^T \left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta } \right) \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathscr {L}}\mathbf {{G}}^{k}\right) \le \frac{{\widetilde{\varepsilon }}^2}{2}\left\| {\mathscr {L}}\mathbf {{U}}^{k}\right\| ^2+ \frac{1}{2{\widetilde{\varepsilon }}^2}\left\| {\mathscr {L}}\mathbf {{G}}^{k}\right\| ^2. \end{aligned}$$

(44)

Noting previous results

$$\begin{aligned} \displaystyle h_1h_2\left( {\mathscr {L}}\mathbf {{U}}^{k}\right) ^T \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {D}}}_{\beta }\right) \left( {\mathscr {L}}\mathbf {{U}}^{k}\right)\le & {} \displaystyle \frac{C_\beta (\gamma ^2-\gamma +2)}{6(b-a)^\beta }\left\| {\mathscr {L}}\mathbf {{U}}^{k}\right\| ^2, \\ \displaystyle h_1h_2\left( {\mathscr {L}}\mathbf {{U}}^{k}\right) ^T\left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta }\right) \left( {{\mathbf {D}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathscr {L}}\mathbf {{U}}^{k}\right)\le & {} \displaystyle \frac{C_\gamma (\beta ^2-\beta +2)}{6(d-c)^\gamma }\left\| {\mathscr {L}}\mathbf {{U}}^{k}\right\| ^2, \\ \displaystyle h_1h_2\left( {\mathscr {L}}\mathbf {{U}}^{k}\right) ^T \left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta } \right) \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathscr {L}}\mathbf {{F}}^{k}\right)\le & {} \frac{{\widetilde{\varepsilon }}^2}{2}\left\| {\mathscr {L}}\mathbf {{U}}^{k}\right\| ^2+ \frac{1}{2{\widetilde{\varepsilon }}^2}\left\| {\mathscr {L}}\mathbf {{F}}^{k}\right\| ^2, \end{aligned}$$

and

$$\begin{aligned}&\displaystyle h_1h_2\left( {\mathscr {L}}\mathbf {{U}}^{k}\right) ^T \left( {{\mathbf {I}}}_{2}\otimes {{\mathbf {A}}}_{\beta }\right) \left( {{\mathbf {A}}}_{\gamma }\otimes {{\mathbf {I}}}_{1}\right) \left( {\mathcal {B}}_{1}^{\alpha ,\lambda }\mathbf {{U}}^{k}\right) \\&\quad \ge \displaystyle \frac{(\beta ^2-\beta +2)(\gamma ^2-\gamma +2)}{72}{\mathcal {B}}_{1}^{\alpha ,2\lambda } \left\| \mathbf {{U}}^{k}\right\| ^2, \end{aligned}$$

and once again using (43) and (44), one can get the following result

$$\begin{aligned} \displaystyle {\mathcal {B}}_{1}^{\alpha ,2\lambda } \left\| \mathbf {{U}}^{k}\right\| ^2 \le \frac{36}{{\widetilde{\varepsilon }}^2(\beta ^2-\beta +2)(\gamma ^2-\gamma +2)}\left( \left\| {\mathscr {L}}\mathbf {{G}}^{k}\right\| ^2+\left\| {\mathscr {L}}\mathbf {{F}}^{k}\right\| ^2\right) , \end{aligned}$$

(45)

where

$$\begin{aligned} \displaystyle {\widetilde{\varepsilon }}^2=\frac{1}{2}\varepsilon ^2= -\frac{1}{6}\left[ \frac{\kappa _\beta C_\beta (\gamma ^2-\gamma +2)}{(b-a)^\beta } +\frac{\kappa _\gamma C_\gamma (\beta ^2-\beta +2)}{(d-c)^\gamma }\right] >0. \end{aligned}$$

Denote ${\mathbf {U}}^k_\lambda =e^{\lambda k\tau }{\mathbf {U}}^k$, then from (45), it leads to that

$$\begin{aligned} \displaystyle q_{0}^{(\alpha )}\left\| \mathbf {{U}}^{k}_\lambda \right\| ^2\le & {} \displaystyle \sum _{\ell =1}^{k-1}\left( q_{k-\ell -1}^{(\alpha )}- q_{k-\ell }^{(\alpha )}\right) \left\| \mathbf {{U}}^{\ell }_\lambda \right\| ^2+q_{k-1}^{(\alpha )}\left[ \left\| \mathbf {{U}}^{0}_\lambda \right\| ^2\right. \nonumber \\&\left. \displaystyle +\, \frac{36T^\alpha e^{2\lambda T}\varGamma (1-\alpha )}{{\widetilde{\varepsilon }}^2(\beta ^2-\beta +2)(\gamma ^2-\gamma +2)} \left( \left\| {\mathscr {L}}\mathbf {{G}}^{k}\right\| ^2+\left\| {\mathscr {L}}\mathbf {{F}}^{k}\right\| ^2\right) \right] . \end{aligned}$$

(46)

Let

$$\begin{aligned} \displaystyle \mathcal {\widetilde{E}}^N=\left\| \mathbf {{U}}^{0}_\lambda \right\| ^2+\frac{36T^\alpha e^{2\lambda T}\varGamma (1-\alpha )}{{\widetilde{\varepsilon }}^2(\beta ^2-\beta +2)(\gamma ^2-\gamma +2)} \max _{1\le k\le N}\left( \left\| {\mathscr {L}}\mathbf {{G}}^{k}\right\| ^2+\left\| {\mathscr {L}}\mathbf {{F}}^{k}\right\| ^2\right) , \end{aligned}$$

(47)

then (46) can be further written as

$$\begin{aligned} \displaystyle q_{0}^{(\alpha )}\left\| \mathbf {{U}}^{k}_\lambda \right\| ^2\le \sum _{\ell =1}^{k-1} \left( q_{k-\ell -1}^{(\alpha )}- q_{k-\ell }^{(\alpha )}\right) \left\| \mathbf {{U}}^{\ell }_\lambda \right\| ^2+ q_{k-1}^{(\alpha )}\mathcal {\widetilde{E}}^N,\quad 1\le k\le N. \end{aligned}$$

(48)

Same as the previous proof process, we can easily get the following result based on (48) and the mathematical induction,

$$\begin{aligned} \displaystyle \left\| \mathbf {{U}}^{k}_\lambda \right\| ^2\le \mathcal {\widetilde{E}}^N,\quad 1\le k\le N. \end{aligned}$$

(49)

Note that $\left\| \mathbf {{U}}^{k}_\lambda \right\| ^2 =\left\| e^{\lambda k\tau }\mathbf {{U}}^{k}\right\| ^2 =e^{2\lambda k\tau }\left\| \mathbf {{U}}^{k}\right\| ^2$, and combined with (47), (48) and (49), resulting in the following desired result

$$\begin{aligned}&\displaystyle \left\| \mathbf {{U}}^{k}\right\| ^2 \\&\quad \le \displaystyle e^{-2\lambda k\tau }\mathcal {\widetilde{E}}^N \\&\quad \le \displaystyle e^{-2\lambda k\tau }\left[ \left\| \mathbf {{U}}^{0}_\lambda \right\| ^2+\frac{36T^\alpha e^{2\lambda T}\varGamma (1-\alpha )}{{\widetilde{\varepsilon }}^2(\beta ^2-\beta +2)(\gamma ^2-\gamma +2)} \max _{1\le k\le N}\left( \left\| {\mathscr {L}}\mathbf {{G}}^{k}\right\| ^2+\left\| {\mathscr {L}}\mathbf {{F}}^{k}\right\| ^2\right) \right] \\&\quad \le \displaystyle \left\| \mathbf {{U}}^{0}_\lambda \right\| ^2+\frac{36T^\alpha e^{2\lambda T}\varGamma (1-\alpha )}{{\widetilde{\varepsilon }}^2(\beta ^2-\beta +2)(\gamma ^2-\gamma +2)}\max _{1\le k\le N}\left( \left\| {\mathscr {L}}\mathbf {{G}}^{k}\right\| ^2+\left\| {\mathscr {L}}\mathbf {{F}}^{k}\right\| ^2\right) \\&\quad \le \displaystyle e^{2\lambda T}\left[ \left\| \mathbf {{U}}^{0}\right\| ^2+\frac{36T^\alpha \varGamma (1-\alpha )}{{\widetilde{\varepsilon }}^2(\beta ^2-\beta +2)(\gamma ^2-\gamma +2)}\max _{1\le k\le N} \left( \left\| {\mathscr {L}}\mathbf {{G}}^{k}\right\| ^2+\left\| {\mathscr {L}}\mathbf {{F}}^{k}\right\| ^2\right) \right] \\&\quad =\displaystyle e^{2\lambda T}\left[ \left\| {\varvec{\Psi }}\right\| ^2+\frac{36T^\alpha \varGamma (1-\alpha )}{{\widetilde{\varepsilon }}^2(\beta ^2-\beta +2)(\gamma ^2-\gamma +2)}\max _{1\le k\le N} \left( \left\| {\mathscr {L}}\mathbf {{G}}^{k}\right\| ^2+\left\| {\mathscr {L}}\mathbf {{F}}^{k}\right\| ^2\right) \right] . \end{aligned}$$