Article Contents
Article Contents

# Locally conservative finite difference schemes for the modified KdV equation

• * Corresponding author: Peter E. Hydon
• Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic nonlinearity.

In principle, a simplified version of the direct approach also works for equations with polynomial nonlinearity of higher degree. For the modified Korteweg-de Vries equation, whose nonlinear term is cubic, this approach yields several new families of second-order accurate schemes that preserve mass and either energy or momentum. Two of these families contain Average Vector Field schemes of the type developed by Quispel and co-workers. Numerical tests show that each family includes schemes that are highly accurate compared to other mass-preserving methods that can be found in the literature.

Mathematics Subject Classification: Primary: 65M06, 37K05; Secondary: 39A14.

 Citation:

• Figure 1.  Example of a rectangular stencil for mKdV. PDEs and conservation laws are preserved to second order at the central point $(x, t)$; densities and fluxes are second-order at $(x, t-\Delta t/2)$ and $(x-\Delta x/2, t)$, respectively

Figure 2.  Two-soliton problem for the mKdV equation. Top: Initial condition (dashed line) and solution of $\mbox{EC}_{10}(0.04)$ with $\Delta x = 0.1$, $\Delta t = 0.025$ at time $T = 10$ (solid line). Bottom: Top of the faster soliton; exact profile (solid line) and solutions of $\mbox{EC}_{10}(0.04)$ (circles), narrow box (squares), multisymplectic (diamonds) and $\mbox{EC}_{10}(0)$ (crosses)

Figure 3.  Two-soliton problem for the mKdV equation with $\Delta x = 0.2$, $\Delta t = 0.05$ at time $T = 10$. Top of the soliton: exact profile (solid line) and solutions of $\mbox{EC}_{10}(0.05)$ (circles), narrow box scheme (squares), multisymplectic scheme (diamonds) and $\mbox{EC}_{10}(0)$ (crosses)

Figure 4.  Breather problem. Top: Numerical solution given by method $\mbox{EC}_{8}(2.22)$ with stepsizes $\Delta x = 0.02$, $\Delta t = 0.002$ at time $T = 0.4$. Bottom: exact profile (solid line) and solutions of methods $\mbox{EC}_{8}(2.22)$ (circles), multisymplectic (diamonds) and $\mbox{EC}_{10}(0)$ (crosses) at the final time (markers at every tenth point)

Table 1.  Errors in conservation laws and solutions for the two-soliton problem for the mKdV equation, with $\Delta x\! = \!0.1$, $\Delta t\! = \!0.025$. An asterisk denotes the error that is minimized

 Method $\text{Err}_1$ $\text{Err}_2$ $\text{Err}_3$ Sol. Err. $\text{Err}_{\phi_1}$ $\text{Err}_{\phi_2}$ $\text{Err}_{\phi}$ $\mbox{EC}_{8}(0)$ 1.74e-13 0.0036 5.13e-13 0.3701 -0.51 -0.06 -0.45 $\mbox{EC}_{8}(1)$ 1.33e-13 0.0732 8.01e-13$^*$ 0.0085$^*$ 0 -0.01 0.01 $\mbox{EC}_{8}(-0.05)$ 6.22e-14$^*$ 1.81e-04$^*$ 4.65e-13 0.3857 -0.53 -0.07 -0.46 $\mbox{MC}_{8}(0)$ 2.13e-13 3.69e-13 0.0632 0.2396 -0.32 -0.04 -0.28 $\mbox{MC}_{8}(-0.077)$ 1.21e-13 3.32e-13 0.0032$^*$ 0.0051$^*$ 0 0.01 -0.01 $\mbox{MC}_{8}(-0.073)$ 6.93e-14 1.46e-13$^*$ 5.55e-04$^*$ 0.0139 -0.02 0.01 -0.03 $\mbox{AVF}_\text{EC}$; $\mbox{EC}_{10}(0)$ 3.91e-14 0.0142 4.80e-14 0.0167 -0.02 0.01 -0.03 $\mbox{EC}_{10}(0.04)$ 3.73e-14 0.0114 9.41e-14$^*$ 0.0030$^*$ 0 0.01 -0.01 $\mbox{EC}_{10}(0.20)$ 5.15e-14$^*$ 1.82e-04$^*$ 5.51e-14 0.0627 0.08 0.02 0.06 $\mbox{AVF}_\text{MC}$; $\mbox{MC}_{10}(0)$ 4.62e-14 5.68e-14 0.0358 0.0756 -0.10 0 -0.10 $\mbox{MC}_{10}(0.19)$ 4.26e-14 5.33e-14 0.0359$^*$ 0.0051$^*$ 0 0.01 -0.01 Narrow box 1.28e-13 0.0117 7.0014 0.0742 0.10 0.02 0.08 Multisymplectic 6.04e-14 0.0058 6.8991 0.2279 -0.31 -0.04 -0.27

Table 2.  Errors in conservation laws and solutions for the two-soliton problem for the mKdV equation, with $\Delta x = 0.2$, $\Delta t = 0.05$. An asterisk denotes the error that is minimized

 Method $\text{Err}_1$ $\text{Err}_2$ $\text{Err}_3$ Sol. Err. $\text{Err}_{\phi_1}$ $\text{Err}_{\phi_2}$ $\text{Err}_{\phi}$ $\mbox{EC}_{8}(0)$ 4.62e-14 0.0155 6.93e-14 0.9599 -1.84 -0.26 -1.58 $\mbox{EC}_{8}(0.97)$ 3.55e-14 0.2754 1.14e-13$^*$ 0.0358$^*$ 0 -0.03 0.03 $\mbox{EC}_{8}(-0.06)$ 4.26e-14$^*$ 5.19e-04$^*$ 1.15e-13 0.9798 -1.93 -0.26 -1.67 $\mbox{MC}_{8}(0)$ 4.44e-14 9.41e-14 0.2363 0.7553 -1.21 -0.15 -1.06 $\mbox{MC}_{8}(-0.079)$ 6.57e-14 1.42e-13 0.0138$^*$ 0.0215$^*$ 0 0.05 -0.05 $\mbox{MC}_{8}(-0.075)$ 4.09e-14 7.11e-14$^*$ 0.0021$^*$ 0.0567 -0.06 0.04 -0.10 $\mbox{AVF}_\text{EC}$; $\mbox{EC}_{10}(0)$ 2.13e-14 0.0574 3.73e-14 0.0725 -0.1 0.02 -0.12 $\mbox{EC}_{10}(0.05)$ 2.66e-14 0.0438 4.09e-14$^*$ 0.0116$^*$ 0 0.03 -0.03 $\mbox{EC}_{10}(0.21)$ 2.13e-14$^*$ 6.74e-04$^*$ 4.97e-14 0.2571 0.35 0.08 0.27 $\mbox{AVF}_\text{MC}$; $\mbox{MC}_{10}(0)$ 1.95e-14 4.80e-14 0.1461 0.2959 -0.40 -0.01 -0.39 $\mbox{MC}_{10}(0.19)$ 2.31e-14 2.49e-14 0.1477$^*$ 0.0205$^*$ 0 0.08 -0.08 Narrow box 4.97e-14 0.0459 6.8421 0.3054 0.40 0.09 0.31 Multisymplectic 2.66e-14 0.0228 6.4635 0.7278 -1.15 -0.17 -0.98

Table 3.  Errors in conservation laws and solution for the breather problem, setting $\Delta x = 0.02$, $\Delta t = 0.002$. An asterisk denotes the error that is minimized

 Method $\text{Err}_1$ $\text{Err}_2$ $\text{Err}_3$ Solution error $\mbox{EC}_{8}(0)$ 6.76e-13 0.1091 1.33e-10 0.9099 $\mbox{EC}_{8}(2.22)$ 2.97e-13 0.3979 1.55e-10$^*$ 0.0144$^*$ $\mbox{EC}_{8}(0.49)$ 9.59e-13$^*$ 0.0079$^*$ 1.05e-10 0.7442 $\mbox{MC}_{8}(0)$ 6.25e-13 5.31e-12 7.534 0.7666 $\mbox{MC}_{8}(-0.165)$ 3.03e-13 2.74e-12 2.3599$^*$ 0.0497$^*$ $\mbox{MC}_{8}(-0.128)$ 7.24e-13 7.60e-12$^*$ 0.1728$^*$ 0.1931 $\mbox{AVF}_\text{EC}$; $\mbox{EC}_{10}(0)$ 3.28e-14 0.1765 1.24e-11 0.4042 $\mbox{EC}_{10}(0.92)$ 3.53e-14 0.0296 2.63e-11$^*$ 0.0295$^*$ $\mbox{EC}_{10}(0.78)$ 5.96e-14$^*$ 0.0095$^*$ 1.35e-11 0.0708 $\mbox{AVF}_\text{MC}$; $\mbox{MC}_{10}(0)$ 1.14e-13 9.24e-13 4.3586 0.5040 $\mbox{MC}_{10}(1.15)$ 5.34e-14 7.03e-13 4.8298$^*$ 0.0219$^*$ Narrow box 1.00e-12 0.0382 566.37 0.3477 Multisymplectic 2.60e-13 0.0184 539.40 0.7994
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