A Convergent Finite Difference Method for a Nonlinear Variational Wave Equation
نویسنده
چکیده
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 = u0 and ut|t=0 = v0. Introducing Riemann invariants R = ut + cux and S = ut − cux, the variational wave equation is equivalent to Rt − cRx = c̃(R2 − S2) and St + cSx = −c̃(R2 − S2) with c̃ = 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 ∈ L1(R)∩L3(R) are nonpositive. The numerical scheme is illustrated on several examples.
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