Unconditional stability and optimal error estimates of a Crank-Nicolson Legendre-Galerkin method for the two-dimensional second-order wave equation

Abstract

This paper presents a fully discrete scheme by discretizing the space with the Legendre-Galerkin method and the time with the Crank-Nicolson method to solve the two-dimensional second-order wave equation. Unconditional stability and optimal error estimates in both L² and H¹ norms of the fully discrete Crank-Nicolson Galerkin method are obtained. Numerical results confirm exponential convergence of the proposed method in space and second-order convergence in time. Also, the numerical experiments show the discrete energy conservation and efficiency of long-time numerical calculation.

Appendix. Proof of Lemma 2.2

Derive the inequality: for $v\in {V_{N}^{2}}$,

$$ 4 \| v\|^{2} \le \mathcal{A}(v,v)\le \frac{1}{2}N(N+1) (N^{2}+3N+3) \| v\|^{2}. $$

(A.1)

Recall that inverse inequality (cf. [15, P. 115]): for $v\in P_{N}(\supseteq V_{N}),$

$$ {\int}^{1}_{-1} \left( \partial_{x} v\right)^{2}\mathrm{d} x \le \left( \sum\limits_{n=0}^{N} n(n+1)(n+1 / 2)\right) {\int}^{1}_{-1}v^{2}\mathrm{d} x. $$

(A.2)

Using the identities

$$ \sum\limits_{n=1}^{N} n =\frac{N(N+1)}{2}, \quad \sum\limits_{n=1}^{N} n^{2} = \frac{N(N+1)(2 N+1)}{6},\quad \sum\limits_{n=1}^{N} n^{3} =\frac{N^{2}(N+1)^{2}}{4}, $$

we get

$$ {\int}^{1}_{-1} \left( \partial_{x} v\right)^{2}\mathrm{d} x \le \frac{1}{4}N(N+1) (N^{2}+3N+3){\int}^{1}_{-1}v^{2}\mathrm{d} x. $$

(A.3)

Together with (2.6), we obtain

$$ \begin{array}{@{}rcl@{}} \mathcal{A}(v,v)&=&\|\nabla u\|^{2}={\int}^{1}_{-1}{\int}^{1}_{-1}\left\{\left( \partial_{x} v\right)^{2}+\left( \partial_{y} u\right)^{2}\right\}\mathrm{d} x \mathrm{d} y\\ &=& {\int}^{1}_{-1}\left\{{\int}^{1}_{-1}\left( \partial_{x} v\right)^{2} \mathrm{d} x \right\}\mathrm{d} y+{\int}^{1}_{-1}\left\{{\int}^{1}_{-1}\left( \partial_{y} v\right)^{2} \mathrm{d} y \right\}\mathrm{d} x\\ &\le & \frac{1}{2}N(N+1) (N^{2}+3N+3) {\int}^{1}_{-1}{\int}^{1}_{-1}v^{2} \mathrm{d} x \mathrm{d} y = \frac{1}{2}N(N+1) (N^{2}+3N+3)\| v\|^{2}. \end{array} $$

Then, exploiting Poincaré inequality: for v ∈ V_N,

$$ \begin{array}{@{}rcl@{}} {\int}^{1}_{-1}v^{2}\mathrm{d} x &=& {\int}^{1}_{-1}\left( {\int}^{x}_{-1}1\cdot \partial_{x} v(s) \mathrm{d} s\right)^{2}\mathrm{d} x \le {\int}^{1}_{-1}\left\{(1+x){\int}^{x}_{-1} \left( \partial_{x} v(s)\right)^{2} \mathrm{d} s\right\} \mathrm{d} x \\ &\le& {\int}^{1}_{-1} \left( \partial_{x} v\right)^{2} \mathrm{d} x{\int}^{1}_{-1}(1+x)\mathrm{d} x =2{\int}^{1}_{-1} \left( \partial_{x} v\right)^{2} \mathrm{d} x. \end{array} $$

(A.4)

Together with (2.6), we obtain

$$ \begin{array}{@{}rcl@{}} \mathcal{A}(v,v) &=&\|\nabla v\|^{2}={\int}^{1}_{-1}{\int}^{1}_{-1}\left\{\left( \partial_{x} v\right)^{2}+\left( \partial_{y} v\right)^{2}\right\}\mathrm{d} x \mathrm{d} y\\ &=& {\int}^{1}_{-1}\left\{{\int}^{1}_{-1}\left( \partial_{x} v\right)^{2} \mathrm{d} x \right\}\mathrm{d} y+{\int}^{1}_{-1}\left\{{\int}^{1}_{-1}\left( \partial_{y} v\right)^{2}\mathrm{d} y \right\}\mathrm{d} x\\ &\ge & 4 {\int}^{1}_{-1}{\int}^{1}_{-1}v^{2} \mathrm{d} x \mathrm{d} y = 4 \| v\|^{2}. \end{array} $$

(A.5)

By Definition 2.1 with v → u, we have

$$ \mathcal{A}(u,u)=\lambda_{\mathcal{A}} (u,u). $$

By (A.1), we obtain

$$ 4 \| u\|^{2}=\lambda_{\mathcal{A}}\| u\|^{2}\le \frac{1}{2}N(N+1) (N^{2}+3N+3) \| u\|^{2}, $$

which implies (2.7) and $\lambda _{\mathcal {A}}\in \mathbb {R}^{+}$. This completes the proof.