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A Local Discontinuous Galerkin method for the modified Camassa--Holm equation
Xiang-Ke Chang, Yong Liu, Qi Tao
TL;DR
The paper addresses numerical discretization of the modified Camassa–Holm equation, whose cubic nonlinearity and high-order derivatives complicate LDG error analysis. It proposes a semi-discrete LDG method with flux choices for conservation or stability, derives an error estimate, and reports sharp convergence and peakon-capturing numerical results.
Problem
Error estimates for LDG methods applied to nonlinear wave equations with higher-order derivatives are difficult, especially because nonlinear analyses commonly require a priori numerical-solution bounds.
Method
The paper develops a unified semi-discrete LDG method for the mCH equation and exploits polynomial nonlinear structure, projections, nonlinear stability, and auxiliary-variable control in its error analysis.
Results
The method establishes semi-discrete energy conservation and stability, proves an a priori error estimate for smooth solutions, and numerically confirms sharp k-th order convergence and peakon capture.
Takeaways & Limitations
The proposed LDG schemes are of arbitrarily high order and can effectively capture peakon solutions of the mCH equation.
Takeaways & Limitations
The analysis remains semi-discrete, and handling nonlinear terms involving the auxiliary variable r requires an a priori bound on r, justified for k > 1.
Abstract
from arXiv · showhide
In this paper, we propose a local discontinuous Galerkin (LDG) method for the modified Camassa-Holm (mCH) equation that contains cubic nonlinearity and high-order derivative terms. Energy conservation and stability are obtained using the conservative and dissipative fluxes, respectively. The error estimate of the semi-discrete scheme is also established. Numerical examples verify that our theoretical findings are sharp and the proposed LDG method is capable of capturing the peakon solution of the mCH equation.
1 Introduction
The paper develops a semi-discrete LDG method for the cubic, high-order modified Camassa–Holm equation, targeting conservation, stability, and optimal error estimates. The analysis uses polynomial nonlinear structure and auxiliary-variable control, while numerical experiments test convergence and peakon capture.
- Motivation: The mCH equation is a cubic-nonlinear, peakon-bearing variant of the Camassa–Holm equation describing unidirectional shallow-water waves.It admits single, periodic, and multipeakon solutions and has infinitely many conservation laws.
- Motivation: LDG extends discontinuous Galerkin methods to high-order equations by rewriting them as equivalent first-order systems.The framework offers localized data structures, parallel scalability, and flexible meshes.
- Contributions: The paper proposes a unified LDG method for the mCH equation with a free real parameter κ, without restricting κ to zero or nonzero values.The method establishes semi-discrete energy conservation with central fluxes and stability with alternating fluxes.
- Contributions: Error analysis is difficult for nonlinear wave equations with higher-order derivatives, and prior LDG analyses commonly use an a priori bound on the numerical solution.The paper avoids an a priori assumption on u by exploiting discrete H1 control through the auxiliary variable r, while retaining an a priori bound on r.
- Results: Numerical experiments corroborate the sharpness of the predicted k-th order convergence and demonstrate effective capture of one-peakon and periodic peakon solutions.The paper presents the discretization, conservation and stability analysis, semi-discrete error analysis, and numerical experiments in successive sections.
2 The LDG scheme for the mCH equation
The semi-discrete LDG scheme rewrites the mCH equation as a first-order system with auxiliary variables and discontinuous polynomial finite elements. Numerical flux choices distinguish dissipative and conservative formulations.
- 2.1 Notations: The computational domain is partitioned into cells, with a quasi-uniform mesh and a discontinuous finite element space of degree at most k.The mesh satisfies hj/h ≥ ρ as h tends to zero, for a positive constant ρ.
- 2.1 Notations: The broken Sobolev space permits functions to be discontinuous across element interfaces, with one-sided traces used at cell boundaries.The construction is based on piecewise polynomial functions over the mesh cells.
- 2.2 The LDG scheme: The LDG formulation introduces auxiliary variables R = Ux, P = (U^2R)x, and S = (1/3R^3)x to convert the high-order equation into a first-order system.The mCH nonlinearities are represented by f(U) = 2κ^2U + U^3 and g(R) = 1/3R^3.
- 2.2 The LDG scheme: The semi-discrete scheme seeks u, p, r, m, and s in the discontinuous finite element space for each time t and against test functions from that space.The numerical solution is initialized as an approximation of U0(x) and uses periodic boundary conditions.
- 2.2 The LDG scheme: Dissipative fluxes use upwind choices for f and g, whereas conservative fluxes use central choices for both fluxes.These fluxes enter the global variational LDG formulation after summing the cellwise equations.
- 2.2 The LDG scheme: The scheme’s global formulation combines the time derivative, nonlinear flux terms, auxiliary-variable bilinear forms, and numerical fluxes.Standard inverse and discrete Sobolev inequalities, together with auxiliary-to-primary variable relationships, support later analysis.
3 The conservation and stability of the LDG scheme
The LDG scheme preserves numerical momentum and supports distinct dissipative and conservative formulations through appropriate numerical flux choices. The resulting energy identities establish the scheme’s conservation and stability properties.
- Analysis: The conservation and stability analysis is conducted for the semi-discrete LDG scheme (2.4).The proof combines discrete equations using selected test functions and the stated lemma.
- Conservation: The numerical scheme is conservative with respect to the numerical momentum density m.This follows by choosing v = 1 in the discrete formulation.
- Dissipative scheme: The dissipative formulation uses numerical fluxes (2.5a) and (2.5b).Its energy behavior is derived from the first and second energy equations of the scheme.
- Conservative scheme: The conservative formulation uses numerical fluxes (2.5a) and (2.5c).The corresponding identities yield the stated conservation relations, including dE/dt = 0.
4 Error estimates of the LDG method
The paper derives semi-discrete LDG error estimates for the mCH equation by decomposing errors, establishing energy identities, and controlling nonlinear terms under smoothness and approximation assumptions. The resulting estimate applies for k ≥1, while Θ2 avoids an a priori error assumption but Θ3 still requires one.
- Error-estimate setup: The error analysis considers the dissipative LDG scheme, with extension to the conservative scheme stated to be direct.The analysis introduces projections, error decompositions, and error equations for the numerical and exact solutions.
- Error-estimate setup: The proof uses L2 and Gauss-Radau projections, whose approximation properties separate high-order projection errors from discrete errors.The total errors are decomposed into projection errors η and discrete errors ξ, enabling subsequent energy estimates.
- Main estimate: For k ≥1, Theorem 4.1 establishes the stated a priori error estimate under the exact-solution smoothness and initial-condition assumptions.The constant in the estimates is independent of h, while Θ3 still requires an a priori assumption because only ∥r∥L2 boundedness is available.
- Nonlinear-term estimates: Θ2 requires no a priori error assumption because polynomial nonlinearity, the u–r relation, energy stability, and L∞ control of u control its higher-order terms.The same relation also controls derivatives of ξu and associated boundary terms.
- Nonlinear-term estimates: The Θ4 estimate handles nonlinear differential and boundary terms through nonlinear stability, projection decomposition, integration by parts, and inverse inequalities.Projection-error terms are grouped separately, while the remaining terms require careful treatment.
5 Numerical experiments
Numerical experiments test the LDG method on smooth soliton, one-peakon, and periodic peakon solutions. The results confirm sharp error estimates, expected convergence, and distinct long-time energy behavior for conservative and dissipative schemes.
- Experimental setup: The experiments use classical fourth-order Runge–Kutta time stepping with the CFL condition Δt = O(h), measuring error in the energy norm.The temporal and spatial step sizes are denoted by Δt and h, respectively.
- Soliton tests: For the smooth soliton with κ = 0, both schemes achieve k-th order convergence for α = 0.5 and 0.7, while α = 0.8 shows oscillatory orders near a singular profile.The limiting soliton profile loses smoothness at the crest as α approaches 3/2 ≃ 0.866, confirming the sharpness of the error estimates.
- Soliton tests: For the κ ≠ 0 one-soliton test, both conservative and dissipative schemes achieve k-th order convergence for α = 0.3, 0.5, and 0.7.The parameters κ = 1 and α = 0.3, 0.5, 0.7 ensure solution smoothness in the tested setting.
- Peakon tests: The conservative LDG method captures the one-peakon structure better than the dissipative method during long-time evolution.For the one-peakon test, the domain is Ω = (−20, 20), the terminal time is T = 5.0, and polynomial degree k = 3 is used.
- Energy behavior: The conservative scheme maintains the energy difference near machine precision, whereas the dissipative scheme dissipates energy monotonically.The same energy-conservation and energy-stability behavior is observed for the periodic peakon test.
- Peakon tests: For the periodic peakon solution, the conservative and dissipative schemes again exhibit energy conservation and energy stability, respectively.The computation uses k = 3 on a uniform mesh with N = 1280 over the periodic domain Ω = (−3π, 3π).
6 Concluding remarks
The proposed LDG method achieves energy stability for general solutions and an a priori error estimate for smooth solutions, while numerical experiments capture peakon solutions.
- The LDG method proves energy stability for general mCH solutions and an a priori error estimate for smooth solutions.
- The nonlinear stability handles nonlinear spatial discretization terms and supports an optimal error estimate for k > 1.
- Numerical results show arbitrarily high-order accuracy and effective capture of peakon solutions.
- The experiments include exact solutions at T = 0 and T = 5, numerical solutions with h = 6.25E-02, and numerical solutions with N = 1280.
- Table 4 reports errors and orders for the dissipative scheme at terminal time T = 5.0.