Fast and High-Order Accuracy Numerical Methods for Time-Dependent Nonlocal Problems in ${\pmb {\mathbb {R}}}^2$

Abstract

In this paper, we study the Crank–Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of ${\mathscr {O}}\left( \tau ^2+h^{4-\gamma }\right)$ with $0<\gamma <1$, where $\tau$ and h are the discretization sizes in the temporal and spatial dimensions respectively. Also we develop the conjugate gradient squared method to solving the resulting discretized nonsymmetric and indefinite systems arising from time-dependent nonlocal problems including two-dimensional cases. By using additive and multiplicative Cauchy kernels in nonlocal problems, structured coefficient matrix-vector multiplication can be performed efficiently in the conjugate gradient squared iteration. Numerical examples are given to illustrate our theoretical results and demonstrate that the computational cost of the proposed method is of $O(M \log M)$ operations where M is the number of collocation points.

Appendices

Appendix A

The matrix ${{\mathscr {G}}}$ in (2.17) consists of four block-structured matrices with Toeplitz-like blocks. Here the block-Toeplitz properties of ${{\mathscr {Q}}}_{\left( M_x-1\right) \times M_x }$, ${{\mathscr {P}}}_{ M_x \times \left( M_x-1\right) }$ and ${{\mathscr {N}}}_{ M_x \times M_x }$ are expressed following:

$$\begin{aligned} {{\mathscr {Q}}}=\left( \begin{array}{ccccccc} {{\mathscr {Q}}}_{1,\frac{1}{2}} &{} {{\mathscr {Q}}}_{1,\frac{3}{2}} &{}{{\mathscr {Q}}}_{1,\frac{5}{2}} &{}{{\mathscr {Q}}}_{1,\frac{7}{2}} &{} \cdots &{}{{\mathscr {Q}}}_{1,M_x-\frac{3}{2}} &{}{{\mathscr {Q}}}_{1,M_x-\frac{1}{2}}\\ {{\mathscr {Q}}}_{2,\frac{1}{2}} &{} {{\mathscr {Q}}}_{1,\frac{1}{2}} &{} {{\mathscr {Q}}}_{1,\frac{3}{2}} &{}{{\mathscr {Q}}}_{1,\frac{5}{2}} &{} \ddots &{}\ddots &{}{{\mathscr {Q}}}_{1,M_x-\frac{3}{2}}\\ {{\mathscr {Q}}}_{3,\frac{1}{2}} &{} {{\mathscr {Q}}}_{2,\frac{1}{2}} &{} {{\mathscr {Q}}}_{1,\frac{1}{2}} &{} {{\mathscr {Q}}}_{1,\frac{3}{2}}&{} \ddots &{}\ddots &{}\vdots \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{}\ddots &{}{{\mathscr {Q}}}_{1,\frac{7}{2}}\\ {{\mathscr {Q}}}_{M_x-2,\frac{1}{2}} &{} \ddots &{} \ddots &{} \ddots &{} \ddots &{}{{\mathscr {Q}}}_{1,\frac{3}{2}} &{}{{\mathscr {Q}}}_{1,\frac{5}{2}}\\ {{\mathscr {Q}}}_{M_x-1,\frac{1}{2}} &{} {{\mathscr {Q}}}_{M_x-2,\frac{1}{2}} &{} \cdots &{} {{\mathscr {Q}}}_{3,\frac{1}{2}} &{} {{\mathscr {Q}}}_{2,\frac{1}{2}} &{}{{\mathscr {Q}}}_{1,\frac{1}{2}} &{}{{\mathscr {Q}}}_{1,\frac{3}{2}} \end{array}\right) _{\left( M_x-1\right) \times M_x }; \end{aligned}$$

and

$$\begin{aligned} {{\mathscr {P}}}=\left( \begin{array}{cccccc} {{\mathscr {P}}}_{\frac{1}{2},1} &{} {{\mathscr {P}}}_{\frac{1}{2},2} &{} {{\mathscr {P}}}_{\frac{1}{2},3}&{} \cdots &{}{{\mathscr {P}}}_{\frac{1}{2},M_x-2} &{}{{\mathscr {P}}}_{\frac{1}{2},M_x-1}\\ {{\mathscr {P}}}_{\frac{3}{2},1} &{} {{\mathscr {P}}}_{\frac{1}{2},1} &{} {{\mathscr {P}}}_{\frac{1}{2},2}&{} \ddots &{}\ddots &{}{{\mathscr {P}}}_{\frac{1}{2},M_x-2}\\ {{\mathscr {P}}}_{\frac{5}{2},1} &{} {{\mathscr {P}}}_{\frac{3}{2},1} &{} {{\mathscr {P}}}_{\frac{1}{2},1}&{} \ddots &{}\ddots &{}\vdots \\ {{\mathscr {P}}}_{\frac{7}{2},1} &{} {{\mathscr {P}}}_{\frac{5}{2},1} &{} {{\mathscr {P}}}_{\frac{3}{2},1}&{} \ddots &{}\ddots &{}{{\mathscr {P}}}_{\frac{1}{2},3}\\ \vdots &{} \ddots &{} \ddots &{}\ddots &{}\ddots &{}{{\mathscr {P}}}_{\frac{1}{2},2}\\ {{\mathscr {P}}}_{M_x-\frac{3}{2},1} &{} \ddots &{} \ddots &{} \ddots &{}{{\mathscr {P}}}_{\frac{3}{2},1} &{}{{\mathscr {P}}}_{\frac{1}{2},1}\\ {{\mathscr {P}}}_{M_x-\frac{1}{2},1} &{}{{\mathscr {P}}}_{M_x-\frac{3}{2},1} &{} \cdots &{}{{\mathscr {P}}}_{\frac{7}{2},1} &{}{{\mathscr {P}}}_{\frac{5}{2},1} &{}{{\mathscr {P}}}_{\frac{3}{2},1} \end{array}\right) _{ M_x \times \left( M_x-1\right) }; \end{aligned}$$

and

$$\begin{aligned} {{\mathscr {N}}}=\left( \begin{array}{ccccc} {{\mathscr {N}}}_{\frac{1}{2},\frac{1}{2}} &{} {{\mathscr {N}}}_{\frac{1}{2},\frac{3}{2}} &{} {{\mathscr {N}}}_{\frac{1}{2},\frac{5}{2}}&{} \cdots &{}{{\mathscr {N}}}_{\frac{1}{2},M_x-\frac{1}{2}}\\ {{\mathscr {N}}}_{\frac{3}{2},\frac{1}{2}} &{} {{\mathscr {N}}}_{\frac{1}{2},\frac{1}{2}} &{} {{\mathscr {N}}}_{\frac{1}{2},\frac{3}{2}}&{} \ddots &{}\vdots \\ {{\mathscr {N}}}_{\frac{5}{2},\frac{1}{2}} &{} {{\mathscr {N}}}_{\frac{3}{2},\frac{1}{2}} &{} {{\mathscr {N}}}_{\frac{1}{2},\frac{1}{2}}&{} \ddots &{}{{\mathscr {N}}}_{\frac{1}{2},\frac{5}{2}}\\ \vdots &{} \ddots &{} \ddots &{}\ddots &{}{{\mathscr {N}}}_{\frac{1}{2},\frac{3}{2}}\\ {{\mathscr {N}}}_{M_x-\frac{1}{2},\frac{1}{2}} &{} \cdots &{} {{\mathscr {N}}}_{\frac{5}{2},\frac{1}{2}}&{} {{\mathscr {N}}}_{\frac{3}{2},\frac{1}{2}} &{}{{\mathscr {N}}}_{\frac{1}{2},\frac{1}{2}} \end{array}\right) _{ M_x \times M_x}. \end{aligned}$$

It should be noted that the above matrices ${{\mathscr {Q}}}$ and ${{\mathscr {P}}}$ have the similar structure properties with ${\mathscr {Q}}$ and ${\mathscr {P}}$ in (2.3).

Appendix B