Article Contents
Article Contents

# Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation

• In this paper, we develop four energy-preserving algorithms for the regularized long wave (RLW) equation. On the one hand, we combine the discrete variational derivative method (DVDM) in time and the modified finite volume method (mFVM) in space to derive a fully implicit energy-preserving scheme and a linear-implicit conservative scheme. On the other hand, based on the (invariant) energy quadratization technique, we first reformulate the RLW equation to an equivalent form with a quadratic energy functional. Then we discretize the reformulated system by the mFVM in space and the linear-implicit Crank-Nicolson method and the leap-frog method in time, respectively, to arrive at two new linear structure-preserving schemes. All proposed fully discrete schemes are proved to preserve the corresponding discrete energy conservation law. The proposed linear energy-preserving schemes not only possess excellent nonlinear stability, but also are very cheap because only one linear equation system needs to be solved at each time step. Numerical experiments are presented to show the energy conservative property and efficiency of the proposed methods.

Mathematics Subject Classification: Primary: 35Q53, 65M08, 65M06.

 Citation:

• Figure 1.  The accuracy of numerical solutions in $L^2$ and $L^{\infty}$ errors of the four schemes with mesh size $\tau = h$

Figure 2.  Comparison of $L^2$ and $L^{\infty}$ errors in numerical solutions and CPU time(s) at $T = 1$, where $c = 1/3$ and $x\in[-40,60]$

Figure 3.  The errors in mass (left) and energy (right) of the four schemes with $c = 1/3$, $\tau = 0.05$, $h = 0.1$ and $x\in[-60,200]$ until $T = 75$

Figure 4.  The evolution of the RLW equation using the scheme LILF with $\sigma = 0.04$ (left), $\sigma = 0.01$ (middle) and $\sigma = 0.001$ (right) at $T = 55$

Figure 5.  The errors in mass (left) and energy (right) of the four schemes with $\sigma = 0.01$ $\tau = 0.05$ and $h = 0.05$ and $x\in[-40,100]$ until $T = 55$

Figure 6.  Initial and undulation profiles with gentle $d = 2$ (top) and $d = 5$ (bottom) at different times using the scheme LILF

Figure 7.  (a) Development of the first undulation from $t = 0$ to $t = 250$ and (b) the behavior of the invariants for $d = 2$ and (c) $d = 5$ by LILF

Table 1.  The invariants and errors of numerical solutions for the scheme FIEP with $c = 0.1$, $\tau = 0.1$, $h = 0.125$ in $[-40,60]$.

 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42983 / / 4 3.97993 0.42983 8.291e-5 3.357e-5 8 3.97993 0.42983 1.633e-4 6.721e-5 12 3.97993 0.42983 2.404e-4 9.791e-5 16 3.97993 0.42983 3.138e-4 1.255e-4

Table 2.  The invariants and errors of numerical solutions for the scheme LIEP with $c = 0.1$, $\tau = 0.1$, $h = 0.125$ in $[-40,60]$.

 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42979 / / 4 3.97993 0.42979 4.020e-5 1.455e-5 8 3.97993 0.42979 8.265e-5 3.124e-5 12 3.97993 0.42979 1.224e-4 4.673e-5 16 3.97993 0.42979 1.614e-4 6.131e-5

Table 3.  The invariants and errors of numerical solutions for the scheme LICN with $c = 0.1$, $\tau = 0.1$, $h = 0.125$ in $[-40,60]$.

 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42983 / / 4 3.97993 0.42983 6.485e-5 2.651e-5 8 3.97993 0.42983 1.270e-4 5.174e-5 12 3.97993 0.42983 1.854e-4 7.413e-5 16 3.97993 0.42983 2.416e-4 9.502e-5

Table 4.  The invariants and errors of numerical solutions for the scheme LILF with $c = 0.1$, $\tau = 0.1$, $h = 0.125$ in $[-40,60]$.

 Time $M$ $H$ $L^2$ error $L^{\infty}$ error Analytical $3.97995$ $0.42983$ / / 0 3.97993 0.42983 / / 4 3.97993 0.42983 1.998e-4 7.882e-5 8 3.97993 0.42983 3.936e-4 1.574e-4 12 3.97993 0.42983 5.842e-4 2.332e-4 16 3.97993 0.42983 7.671e-4 3.021e-4

Table 5.  Numerical comparison at $T = 10$ with $c = 0.1$, $\tau = 0.1$ and $-40\leq x\leq 60$.

 Method $h=0.125$ $h=0.0625$ $L^2$ error $L^{\infty}$ error CPU(s) $L^2$ error $L^{\infty}$ error CPU(s) FIEP 2.023e-4 8.298e-5 1.398 1.363e-4 5.520e-5 1.796 LIEP 1.035e-4 3.946e-5 0.993 1.687e-4 6.716e-5 1.268 LICN 1.566e-4 6.316e-5 1.073 9.172e-5 3.545e-5 1.076 LILF 4.896e-4 1.962e-4 1.006 4.241e-4 1.684e-4 1.128 NC-II [34] 2.088e-4 7.532e-5 1.755 1.234e-4 4.236e-5 2.137 AMC-CN [44] 3.944e-4 1.581e-4 1.506 1.828e-4 7.307e-5 1.988 Linear-CN [6] 2.648e-4 1.088e-4 1.046 7.945e-4 2.957e-4 1.108
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