High-order compact difference schemes based on the local one-dimensional method for high-dimensional nonlinear wave equations

Abstract

In this paper, two compact difference schemes are established for solving two-dimensional (2D) and three-dimensional (3D) nonlinear wave equations with variable coefficients, respectively, by using the local one-dimensional (LOD) method and the fourth-order compact difference approximation formulas of the second-order derivatives. Firstly, a four-step fourth-order compact scheme is derived to solve the 2D nonlinear wave equation. The stability of the scheme 2for solving the linear equation is analyzed by the discrete Fourier method, which shows that it is conditionally stable. Then, the method is extend to solve the 3D nonlinear wave equation and stability condition for the linear equation is also analyzed. Finally, numerical experiments are conducted to verify the accuracy and stability of the proposed schemes.

Appendices

Appendix A

Lemma 1

[36] The sufficient and necessary condition for the roots of the quadratic equation ${\delta ^2} - {b_1}\delta - {c_1} = 0$ with real coefficients to be less than or equal to 1 is $|{{c_1}} |\le 1,|{{b_1}} |\le 1-c_1$.

Theorem 2

The scheme is stable if

$$\begin{aligned} \mathop {\max }\limits _{1 \le i,j,k \le N} |{\frac{{v_{i,j,k} \cdot \tau }}{h}}|= v_{\max } \lambda \le 0.7385, \end{aligned}$$

in which, ${v_{\max }} = \mathop {\max }\limits _{1 \le i,j,k \le N} |{{v_{i,j,k}}} |$.

Proof

Letting $u_{i,j,k}^n = {\eta ^n}{e^{I{\sigma _1}{x_i}}}{e^{I{\sigma _2}{y_j}}}{e^{I{\sigma _3}{z_k}}}$, $u_{i,j,k}^{n + \frac{1}{3}} = {\eta ^{n + \frac{1}{3}}}{e^{I{\sigma _1}{x_i}}}{e^{I{\sigma _2}{y_j}}}{e^{I{\sigma _3}{z_k}}}$, $u_{i,j,k}^{n - \frac{1}{3}} = {\eta ^{n - \frac{1}{3}}}{e^{I{\sigma _1}{x_i}}}{e^{I{\sigma _2}{y_j}}}{e^{I{\sigma _3}{z_k}}}$, $\mathop {\max }\limits _{1 \le i,j,k \le N} v_{i,j,k}^2 = a$, and multiplying by a on the both sides of Eq. (45), we have

$$\begin{aligned}{} & {} \left( {\frac{5}{6} + \frac{{{a}{\lambda ^2}}}{{18}}} \right) {\eta ^{n + \frac{1}{3}}}{e^{I{\sigma _1}{x_i}}}{e^{I{\sigma _2}{y_j}}}{e^{I{\sigma _3}{z_k}}}\nonumber \\{} & {} + \left( {\frac{1}{{12}} - \frac{{{a}{\lambda ^2}}}{{36}}} \right) {\eta ^{n + \frac{1}{3}}}\left( {{e^{I{\sigma _1}{x_{i + 1}}}} + {e^{I{\sigma _1}{x_{i - 1}}}}} \right) {e^{I{\sigma _2}{y_j}}}{e^{I{\sigma _3}{z_k}}} \nonumber \\= & {} \left( {\frac{5}{3} - \frac{{5{a}{\lambda ^2}}}{9}} \right) {\eta ^n}{e^{I{\sigma _1}{x_i}}}{e^{I{\sigma _2}{y_j}}}{e^{I{\sigma _3}{z_k}}}\nonumber \\{} & {} + \left( {\frac{1}{6} + \frac{{5{a}{\lambda ^2}}}{{18}}} \right) {\eta ^n}\left( {{e^{I{\sigma _1}{x_{i + 1}}}} + {e^{I{\sigma _1}{x_{i - 1}}}}} \right) {e^{I{\sigma _2}{y_j}}}{e^{I{\sigma _3}{z_k}}}\nonumber \\{} & {} - \left( {\frac{5}{6} + \frac{{{a}{\lambda ^2}}}{{18}}} \right) {\eta ^{n - \frac{1}{3}}}{e^{I{\sigma _1}{x_i}}}{e^{I{\sigma _2}{y_j}}}{e^{I{\sigma _3}{z_k}}}\nonumber \\{} & {} \!-\! \left( {\frac{1}{{12}} \!-\! \frac{{{a}{\lambda ^2}}}{{36}}} \right) {\eta ^{n - \frac{1}{3}}}\left( {{e^{I{\sigma _1}{x_{i + 1}}}} + {e^{I{\sigma _1}{x_{i - 1}}}}} \right) {e^{I{\sigma _2}{y_j}}}{e^{I{\sigma _3}{z_k}}}.\nonumber \\ \end{aligned}$$

(A-1)

By ${e^{ \pm I\sigma h}} = \cos \sigma h \pm I\sin \sigma h$, we get

$$\begin{aligned}{} & {} \left( {\frac{5}{6} + \frac{{a{\lambda ^2}}}{{18}}} \right) {\eta ^{n + \frac{1}{3}}} + 2\cos {\sigma _1}h\left( {\frac{1}{{12}} - \frac{{a{\lambda ^2}}}{{36}}} \right) {\eta ^{n + \frac{1}{3}}}\nonumber \\= & {} \left( {\frac{5}{3} - \frac{{5a{\lambda ^2}}}{9}} \right) {\eta ^n} + 2\cos {\sigma _1}h\left( {\frac{1}{6} + \frac{{5a{\lambda ^2}}}{{18}}} \right) {\eta ^n}\nonumber \\{} & {} - \left( {\frac{5}{6} + \frac{{a{\lambda ^2}}}{{18}}} \right) {\eta ^{n - \frac{1}{3}}} - 2\cos {\sigma _1}h\left( {\frac{1}{{12}} - \frac{{a{\lambda ^2}}}{{36}}} \right) {\eta ^{n - \frac{1}{3}}}.\nonumber \\ \end{aligned}$$

(A-2)

Letting ${\varepsilon ^{n + \frac{1}{3}}} = {\eta ^n},\,\,{\varepsilon ^n} = {\eta ^{n - \frac{1}{3}}}$, Eq. (A-2) is written in matrix form

$$\begin{aligned} \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {\frac{5}{6}\! \!+ \!\!\frac{{a{\lambda ^2}}}{{18}}\!\! +\!\! \left( {\frac{1}{6} \!-\! \frac{{a{\lambda ^2}}}{{18}}} \right) \cos {\sigma _1}h}&{}\,\,\,\,\,\,0\\ 0&{}\,\,\,\,\,\,1 \end{array}} \right] \left[ \begin{array}{l} {\eta ^{n + \frac{1}{3}}}\\ {\varepsilon ^{n + \frac{1}{3}}} \end{array} \right] \\ = \left[ \! {\begin{array}{*{20}{c}} {\frac{5}{3} \!\!- \!\!\frac{{5a{\lambda ^2}}}{9} \!\!+\!\! \left( {\frac{1}{3} \!\!+ \!\!\frac{{5a{\lambda ^2}}}{9}} \right) \cos {\sigma _1}h}&{}{ \,\,\,\,\,\,-\! \left( {\frac{5}{6} \!\!+\! \!\frac{{a{\lambda ^2}}}{{18}}} \right) \!\!-\! \!\left( {\frac{1}{6}\!\! -\!\! \frac{{a{\lambda ^2}}}{{18}}} \right) \cos {\sigma _1}h}\\ 1&{}\,\,\,\,\,\,0 \end{array}} \!\right] \left[ \begin{array}{l} {\eta ^n}\\ {\varepsilon ^n} \end{array} \right] . \end{array} \end{aligned}$$

(A-3)

Letting ${U^n} = {\left( {{\eta ^n},{\varepsilon ^n}} \right) ^T}$ and substituting it into Eq. (A-3) to get

$$\begin{aligned} \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {\frac{5}{6}\!\! +\!\! \frac{{a{\lambda ^2}}}{{18}} \!\!+\!\! \left( {\frac{1}{6} \!\!- \!\!\frac{{a{\lambda ^2}}}{{18}}} \right) \cos {\sigma _1}h}&{}\,\,\,\,\,\,0\\ 0&{}\,\,\,\,\,\,1 \end{array}} \right] {U^{n + \frac{1}{3}}}\\ = \left[ \!{\begin{array}{*{20}{c}} {\frac{5}{3} \!\!-\!\! \frac{{5a{\lambda ^2}}}{9}\! \!+ \!\!\left( {\frac{1}{3}\!\! +\!\! \frac{{5a{\lambda ^2}}}{9}} \right) \cos {\sigma _1}h}&{}{ \,\,\,\,\,\,- \!\left( {\frac{5}{6} \!\!+\! \!\frac{{a{\lambda ^2}}}{{18}}} \right) \! \!- \!\!\left( {\frac{1}{6}\!\! -\! \!\frac{{a{\lambda ^2}}}{{18}}} \right) \cos {\sigma _1}h}\\ 1&{}\,\,\,\,\,\,0 \end{array}}\! \right] {U^n}. \end{array} \end{aligned}$$

(A-4)

Similarly, Eqs. (46) and (47) can be treated as

$$\begin{aligned} \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {\frac{5}{6} \!+ \!\frac{{a{\lambda ^2}}}{{18}} \!+\! \left( {\frac{1}{6}\! - \!\frac{{a{\lambda ^2}}}{{18}}} \right) \cos {\sigma _2}h}&{}\,\,\,\,\,\,0\\ 0&{}\,\,\,\,\,\,1 \end{array}} \right] {U^{n + \frac{2}{3}}}\\ = \left[ \!{\begin{array}{*{20}{c}} {\frac{5}{3} \!\!-\!\! \frac{{5a{\lambda ^2}}}{9}\!\! +\!\! \left( {\frac{1}{3} \!\!+\!\! \frac{{5a{\lambda ^2}}}{9}} \right) \cos {\sigma _2}h}&{}{ \,\,\,\,\,\,-\! \left( {\frac{5}{6} \!\!+ \!\!\frac{{a{\lambda ^2}}}{{18}}} \right) \!\! - \!\!\left( {\frac{1}{6} \!\!-\!\! \frac{{a{\lambda ^2}}}{{18}}} \!\right) \cos {\sigma _2}h}\\ 1&{}\,\,\,\,\,\,0 \end{array}}\! \right] {U^{n + \frac{1}{3}}}, \end{array} \end{aligned}$$

(A-5)

and

$$\begin{aligned} \begin{array}{l} \left[ {\begin{array}{*{20}{c}} {\frac{5}{6}\!\! +\!\! \frac{{a{\lambda ^2}}}{{18}} \!\!+\!\! \left( {\frac{1}{6} \!\!-\!\! \frac{{a{\lambda ^2}}}{{18}}} \right) \cos {\sigma _3}h}&{}\,\,\,\,\,\,0\\ 0&{}\,\,\,\,\,\,1 \end{array}} \right] {U^{n + 1}}\\ = \left[ \!{\begin{array}{*{20}{c}} {\frac{5}{3}\! \!-\! \!\frac{{5a{\lambda ^2}}}{9}\!\!+ \!\!\left( \!{\frac{1}{3} \!\!+\! \!\frac{{5a{\lambda ^2}}}{9}} \!\right) \cos {\sigma _3}h}&{}{\,\,\,\,\,\,-\! \!\left( \! {\frac{5}{6}\! \!+ \!\!\frac{{a{\lambda ^2}}}{{18}}} \!\right) \! \!-\! \!\left( \! {\frac{1}{6} \!\!- \!\!\frac{{a{\lambda ^2}}}{{18}}} \!\right) \cos {\sigma _3}h}\\ 1&{}\,\,\,\,\,\,0 \end{array}} \!\right] {U^{n + \frac{2}{3}}}. \end{array} \end{aligned}$$

(A-6)

Substituting Eqs. (A-4)-(A-5) into Eq. (A-6), the error propagation matrix is

$$\begin{aligned} G =\left[ {\begin{array}{*{20}{c}} { -\! \frac{{{B_x}}}{{{A_x}}} \!- \!\frac{{{B_z}}}{{{A_z}}} \!+\! \frac{{{B_x}{B_y}{B_z}}}{{{A_x}{A_y}{A_z}}}}&{}\,\,\,\,\,\,{1 \!- \!\frac{{{B_y}{B_z}}}{{{A_y}{A_z}}}}\\ {\frac{{{B_x}{B_y}}}{{{A_x}{A_y}}} \!- \!1}&{}\,\,\,\,\,\,{ -\! \frac{{{B_y}}}{{{A_y}}}} \end{array}} \right] , \end{aligned}$$

in which,

$$\begin{aligned} {A_x}= & {} \frac{5}{6} + \frac{{{a}{\lambda ^2}}}{{18}} + \left( {\frac{1}{6} - \frac{{{a}{\lambda ^2}}}{{18}}} \right) \cos {\sigma _1}h,\\ {A_y}= & {} {\frac{5}{6} + \frac{{{a^2}{\lambda ^2}}}{{18}} + \left( {\frac{1}{6} - \frac{{a{\lambda ^2}}}{{18}}} \right) \cos {\sigma _2}h},\\ {A_z}= & {} {\frac{5}{6} + \frac{{{a}{\lambda ^2}}}{{18}} + \left( {\frac{1}{6} - \frac{{{a}{\lambda ^2}}}{{18}}} \right) \cos {\sigma _3}h},\\ {B_x}= & {} \frac{5}{3} - \frac{{5{a}{\lambda ^2}}}{9} + \left( {\frac{1}{3} + \frac{{5{a}{\lambda ^2}}}{9}} \right) \cos {\sigma _1}h,\\ {B_y}= & {} {\frac{5}{3} - \frac{{5{a}{\lambda ^2}}}{9} + \left( {\frac{1}{3} + \frac{{5{a}{\lambda ^2}}}{9}} \right) \cos {\sigma _2}h},\\ {B_z}= & {} {\frac{5}{3} - \frac{{5{a}{\lambda ^2}}}{9} + \left( {\frac{1}{3} + \frac{{5{a}{\lambda ^2}}}{9}} \right) \cos {\sigma _3}h}. \end{aligned}$$

The characteristic equation can be obtained by

$$\begin{aligned} |{\mu I \!-\! G} |\! = \!{\mu ^2} \!+\! \left( {\frac{{{B_x}}}{{{A_x}}} \!+\! \frac{{{B_y}}}{{{A_y}}} \!+ \frac{{{B_z}}}{{{A_z}}} \!- \frac{{{B_x}{B_y}{B_z}}}{{{A_x}{A_y}{A_z}}}} \right) \mu + 1=0. \end{aligned}$$

According to the Lemma 1, when $|b_1 |= |\frac{{{B_x}{B_y}{B_z}}}{{{A_x}{A_y}{A_z}}}$ $- \left( {\frac{{{B_x}}}{{{A_x}}} + \frac{{{B_y}}}{{{A_y}}} + \frac{{{B_z}}}{{{A_z}}}} \right) |\le 2$, the scheme is stable.

Because $\frac{{{B_x}}}{{{A_x}}}$, $\frac{{{B_y}}}{{{A_y}}}$ and $\frac{{{B_z}}}{{{A_z}}}$ have the same range of values, thus, either $- 2 \le \frac{{{B_x}}}{{{A_x}}},\frac{{{B_y}}}{{{A_y}}},\frac{{{B_z}}}{{{A_z}}} \le - 1$ or $- 1 \le \frac{{{B_x}}}{{{A_x}}},\frac{{{B_y}}}{{{A_y}}},\frac{{{B_z}}}{{{A_z}}} \le 1$ or $1 \le \frac{{{B_x}}}{{{A_x}}},\frac{{{B_y}}}{{{A_y}}},\frac{{{B_z}}}{{{A_z}}}\le 2$, the scheme is stable. Here, we only analyze the value range of $\frac{{{B_x}}}{{{A_x}}}$.

Letting $\cos {\sigma _1}h = \theta ,\theta \in \left[ { - 1,1} \right]$, we assume that

$$\begin{aligned} G(\theta ) = \frac{{{B_x}}}{{{A_x}}} = \frac{{\frac{1}{3}\left( {5 + \theta } \right) + \frac{{5{a}{\lambda ^2}}}{9}\left( {\theta - 1} \right) }}{{\frac{1}{6}\left( {5 + \theta } \right) + \frac{{{a}{\lambda ^2}}}{{18}}\left( {1 - \theta } \right) }}, \end{aligned}$$

$G\left( \theta \right)$ is an increasing function, the value range of the function $G\left( \theta \right)$ is $\left[ {\frac{{12 - 10{a}{\lambda ^2}}}{{6 + {a}{\lambda ^2}}},2} \right]$. Due to either $- 2 \le G\left( \theta \right) \le - 1$ or $- 1 \le G\left( \theta \right) \le 1$ or $1 \le G\left( \theta \right) \le 2$, the scheme is stable. So, when $1 \le \frac{{12 - 10{a}{\lambda ^2}}}{{6 + {a}{\lambda ^2}}} \le 2$, i.e., $a{\lambda ^2} \le \frac{6}{{11}}$, the scheme is stable.

In summary, when $a{\lambda ^2} \le \frac{6}{{11}}$, i.e., ${v_{\max }}\lambda = \sqrt{a} \lambda \le 0.7385$, the scheme is stable.$\square$

Appendix B

Table 10 The ${\Vert {{e_h}} \Vert _\infty }$ for various $\tau$ at $h = \pi /200$ at $T = 1$ for Problem 1

Table 11 The ${\Vert {{e_h}} \Vert _\infty }$ for various T and $\tau$ with $h = \pi /200$ by the HOC-LOD scheme for Problem 1

Table 12 The ${\Vert {{e_h}} \Vert _\infty }$ for different T and $\tau$ at $h = \pi /80$ by the HOC-LOD scheme for Problem 4

Table 13 The ${\Vert {{e_h}}\Vert _\infty }$ and ${\Vert {{e_h}} \Vert _2}$ when $T=0.1,\tau =0.001$ with various h by the HOC-LOD scheme for Problem 5

Table 14 The ${\Vert {{e_h}}\Vert _\infty }$ and ${\Vert {{e_h}} \Vert _2}$ when $T=1,h=2\tau$ with various $\tau$ by the HOC-LOD scheme for Problem 5