New Finite-Volume Relaxation Methods for the Third-Order Differential Equations (Communications in Computational Physics, 2008)
We propose a new method for numerical solution of the third-order differential equations. The key idea is to use relaxation approximation to transform the nonlinear third-order differential equation to a semilinear second-order differential systemwith a source term and a relaxation parameter. The relaxation system has linear characteristic variables and can be numerically solved without relying on Riemann problemsolvers or linear iterations. A non-oscillatory finite volume method for the relaxationsystem is developed. The method is uniformly accurate for all relaxation rates. Numerical results are shown for some nonlinear problems such as the Korteweg-de Viresequation. Our method demonstrated the capability of accurately capturing solitonwave phenomena.