Locally conservative finite difference schemes for the modified KdV equation

Gianluca Frasca-Caccia; Peter E. Hydon

doi:10.3934/jcd.2019015

Article Contents

2019, Volume 6, Issue 2: 307-323. Doi: 10.3934/jcd.2019015

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Locally conservative finite difference schemes for the modified KdV equation

Gianluca Frasca-Caccia and
Peter E. Hydon^,

School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7FS, UK

^* Corresponding author: Peter E. Hydon
^* Corresponding author: Peter E. Hydon

Received: February 28, 2019

Revised: June 30, 2019

Published: November 2019

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Abstract

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.

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

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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

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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)

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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)

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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)

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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

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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

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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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