Arbitrarily accurate non-oscillatory schemes for a non-linear conservation law

Abstract

Arbitrary accuracy DErivative Riemann problem (ADER) approach for constructing nonoscillatory explicit one-step schemes with very high order of accuracy in space and time has been extended to nonlinear scalar conservation laws. The extension of the ADER from a linear equation to a nonlinear equation has choice of two methods: one is based on the state-series expansion and another is based on the direct expansion for the flux function. Regarding the direct expansion, three methods have been presented to replace time derivatives of flux with space derivatives of state and flux. The accuracy of methods based on the direct expansion has been analyzed, together with the discussion on how to increase the convergence rate. By using the presented methods it is possible to design arbitrarily accurate non-oscillatory schemes in principle. Numerical verification for the ADER up to the 5-th order of accuracy showed that the expected convergence rate has been achieved, and advantages compared with the ENO and WENO schemes have been discussed.