A convergent finite-difference method for a nonlinear variational wave equation

Abstract

Abstract. We establish rigorously convergence of a semi-discrete upwind scheme for the nonlinear variational wave equation utt − c(u)(c(u)ux)x = 0 with u| t=0 = u 0 and ut| t=0 = v 0 . Introducing Riemann invariants R = ut + cux and S = ut − cux, the variational wave equation is equivalent to Rt − cRx =c(R 2 − S 2 ) and St + cSx = −c(R 2 − S 2 ) withc = c /(4c). An upwind scheme is defined for this system. We assume that the the speed c is positive, increasing and both c and its derivative are bounded away from zero and that R| t=0 , S| t=0 ∈ L 1 (R) ∩ L 3 (R) are nonpositive. The numerical scheme is illustrated on several examples.

An Energy Conserving Local Discontinuous Galerkin Method for a Nonlinear Variational Wave Equation

“…7-8 show the numerical solutions of the auxiliary variable q = c(u)u x at times t = 0, t = 5, t = 6 and t = 15 for the conservative and dissipative LDG schemes, respectively. The results are consistent with those reported in [3,22]. It is reported that despite the initial data is smooth, the solution develops a singularity in u x around t=6.…”

“…This problem has been tested numerically in [3,17,22]. The exact solution is not available, we use a reference solution computed on a finer mesh with N = 20000.…”

“…For the nonlinear second order wave equation (1.1), a semi-discrete finite difference scheme was first considered in [22], where the authors were able to prove convergence of the generated numerical solution to the dissipative solution, for some restricted choice of c(u). Further in [25] finite difference schemes for approximating the variational wave equation in both one and two space dimensions were developed.…”

Finite difference schemes for the two‐dimensional semilinear wave equation

“…Different efficient and accurate numerical approximations including finite element methods have used continuous Galerkin methods [31][32][33][34][35][36], discontinuous Galerkin methods [37][38][39][40], and mixed finite element methods [41][42][43][44]. In recent years, much attention has been paid to the finite difference schemes for wave equation [45][46][47][48][49]. In Refs.…”

Conservative solutions to a one-dimensional nonlinear variational wave equation

“…Holden and Raynaud [14] have shown that it possesses a global semigroup for conservative weak solutions. A numerical method for studying (1.4) was provided by Holden et al [13]. Recently, Zhang and Zheng [30,31] have proved the global existence of conservative weak solutions for a one-dimensional system of variational wave equations, which is derived from (1.3) by considering the three-dimensional deformations of the director field that depend on a single space variable [2].…”