High-Order Non-Oscillatory Schemes for Nonlinear Conservation Laws with Source Terms

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In this study, the ADER (Arbitrary-Accuracy Derivative Riemann problem) approach for constructing non-oscillatory explicit one-step schemes with very high order of accuracy in space and time has been extended to scalar nonlinear conservation laws with source terms, and further applied to the fluid dynamics problems including source terms so as to develop shock capturing schemes having both high accuracy and high stability.From the view point of keeping the high accuracy, two methods based on the state-series expansion and the direct expansion were shown in the extension of ADER approach from a linear equation to a nonlinear equation. The ADER schemes thus constituted were verified with the Burgers' equation (convex-flux equation) for two test problems of rapidly growing waves and very long-time propagating waves. As results, it was confirmed that the ADER schemes achieve the designed order of accuracy and capture shock and expansion waves clearly Furthermore, for extensive problems wit … More h a linear flux, nonlinear convex fluxes, nonlinear non-convex fluxes numerical verification showed that the ADER schemes achieve the high accuracy, high stability, and non-oscillatory property Then the ADER schemes have been applied to the fluid dynamics problems : first, to the one-dimensional Euler equation system, and next to the shallow water equation system as a source-term problem.We have conducted theoretical analysis on stability as well, observing the difference between properties of the continuous and discrete models. It was discovered that very small errors from the discrete computation may be amplified by some property of discrete model that does not come from the continuous model i.e. the original PDE. It seems that this phenomenon may happen especially in the case of nonlinear hyperbolic conservation laws including linearly degenerate fields. Such conservation laws generally occur in describing the wave propagation phenomena over continuum. This kind of numerical behavior deteriorates the precise estimate of effect from source terms and is important from the viewpoint of reliability of numerical computation. Less