An Efficient Compact Difference Scheme for the Symmetric Regularized Long Wave Equation
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摘要: 为了求出对称正则长波(symmetric regularized long wave, SRLW)方程的数值解，构造了一种新的高效紧致有限差分格式.采用经典的Crank-Nicolson(C-N)格式和外推技术对时间方向一阶导数进行离散化，使用四阶Padé方法和逆紧致算子分别对空间方向一阶和二阶导数进行离散化，使得构造的格式具有线性、非耦合和紧致的特点，极大地提高了求解效率.此外，还对新格式进行了守恒律、先验估计、稳定性、收敛性分析，证明了其在时间上达到二阶、在空间上达到四阶收敛精度.最后，通过一个数值算例验证了理论的正确性和格式的高效性.Abstract: A new efficient and compact finite difference scheme was constructed to obtain numerical solutions of the symmetric regularized long wave equation. The classic Crank-Nicolson (C-N) scheme and the extrapolation technique were used for discretization of the 1st-order derivatives in the temporal direction, the 4th-order Padé method and the inverse compact operator were applied for discretization of the 1st-order and 2nd-order derivatives in the spatial direction, respectively. The constructed scheme has the linear, uncoupled, and compact features, greatly enhancing the computational efficiency. Additionally, analyses on conservation laws, a priori estimates, stability and convergence were conducted for the new scheme, to prove the 2nd-order temporal and the 4th-order spatial convergence accuracies. Finally, the theoretical correctness and efficiency of the scheme were verified through a numerical example.
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Key words:
- symmetric regularized long wave equation /
- finite difference /
- efficient /
- compact
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图 1 u(x, t)和ρ(x, t)的三维效果
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 1. The 3D images of u(x, t) and ρ(x, t)
图 2 格式一与本文格式数值解误差比较
Figure 2. Comparison of errors between scheme 1 and the present scheme
图 3 格式一与本文格式CPU时间比较
Figure 3. Comparison of CPU times between scheme 1 and the present scheme
图 4 h=0.1, τ=0.01时，u(x, t)和ρ(x, t)长时间的波形
Figure 4. Waveforms of u(x, t) and ρ(x, t) for a long time with h=0.1, τ=0.01
表 1 本文格式与格式一的误差、收敛阶和CPU时间
Table 1. The errors, convergence rates and CPU times of the present scheme and scheme 1
(h, τ) present scheme scheme 1 e(h, τ) uratex η(h, τ) ρratex tCPU/s e(h, τ) η(h, τ) tCPU/s τ=h2 $ \left(\frac{1}{4}, \frac{1}{16}\right)$ 1.263 4×10-2 - 1.704 9×10-2 - 1.03 3.937 0×10-3 5.587 9×10-3 9.57 $ \left(\frac{1}{8}, \frac{1}{64}\right)$ 7.852 2×10-4 4.008 0 1.052 8×10-3 4.017 3 13.15 2.468 7×10-4 3.409 6×10-4 142.86 $\left(\frac{1}{16}, \frac{1}{256}\right) $ 4.911 7×10-5 3.998 8 6.565 7×10-5 4.003 1 257.69 1.545 2×10-5 2.127 9×10-5 2 956.44 h=τ $\left(\frac{1}{8}, \frac{1}{8}\right) $ 5.175 3×10-2 - 7.033 0×10-2 - 2.24 1.434 9×10-2 2.166 6×10-2 23.59 $ \left(\frac{1}{16}, \frac{1}{16}\right)$ 1.269 2×10-2 2.027 7 1.712 9×10-2 2.037 6 15.25 3.510 0×10-3 5.024 3×10-3 206.61 $\left(\frac{1}{32}, \frac{1}{32}\right) $ 3.160 4×10-3 2.005 7 4.245 4×10-3 2.012 4 159.42 8.787 4×10-4 1.226 7×10-3 2 472.98 
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表 2 h=0.1, τ=0.01时不同时刻离散质量和能量变化情况
Table 2. Discrete mass and energy changes at different moments with h=0.1, τ=0.01
t Q1n Q2n En 0 13.416 407 864 998 76 8.944 271 909 999 150 34.783 278 796 685 07 2 13.416 407 864 975 92 8.944 271 909 999 168 34.783 287 701 975 17 4 13.416 407 864 952 86 8.944 271 909 999 157 34.783 296 625 146 47 6 13.416 407 864 928 57 8.944 271 909 999 122 34.783 305 535 423 96 8 13.416 407 864 892 85 8.944 271 909 998 717 34.783 314 433 168 47 10 13.416 407 864 750 97 8.944 271 909 994 891 34.783 323 323 411 79
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