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

作者：Holden H^*; Karlsen K H; Risebro N H

来源：IMA Journal of Numerical Analysis, 2009, 29(3): 539-572.

DOI：10.1093/imanum/drn026

• variational wave equation
• convergence of finite-difference schemes
• liquid crystals

We establish rigorously convergence of a semidiscrete upwind scheme for the nonlinear variational wave equation u(tt) - c(u)(c(u)u(x))(x) = 0 with u vertical bar(t= 0) = u(0) and u(t)vertical bar(t=0) = upsilon(0). Introducing Riemann invariants R = u(t) + cu(x) and S = u(t) - cu(x), the variational wave equation is equivalent to R(t) - cR(x) = (c) over tilde (R(2) - S(2)) and S(t) + cS(x) = -(c) over tilde (R(2) - S(2)) with (c) over tilde = c'/(4c). An upwind scheme is defined for this system. We assume that the speed c is positive, increasing and both c and its derivative are bounded away from zero and that R vertical bar(t=0), S vertical bar(t=0) is an element of L(1)(R) boolean AND L(3)(R) are nonpositive. The numerical scheme is illustrated on several examples.

• 出版日期2009-7