Hyperbolic Conservation Laws Of High Order Upwind Scheme With Adaptive Artificial Viscosity

Many problems in nature (such as shallow water) could be described by hyperbolic conservation laws. For this kind of equation, most questions have been still unable to obtain exact solutions in addition to a few simple case, and only solved by numerical simulation with numerical methods. Therefore, the study of numerical methods of hyperbolic conversation laws is one of the important research directions of scientific computing and computational fluid dynamics.In this paper, we discuss high-order upwind scheme with adaptive artificial viscosity term for hyperbolic conversation laws, conclusions are as follows:(1) A high-order upwind scheme is developed for hyperbolic system of conservation laws in the hope of saving computational cost with the help of adaptive artificial viscosity term replacing the role of nonlinear weight. Extensive numerical results are presented to demonstrate the performance of the resulting scheme in terms of essentially non-oscillatory property near discontinuities as well as high efficiency.(2) A high-order upwind scheme with adaptive artificial viscosity term is developed for a pollutant transport model in shallow water. The computational cost of the original weighted essentially non-oscillatory (WENO) scheme is very high due to the computation of the nonlinear weight. The viscosity term is introduced to replace the role of the nonlinear weight to ensure the nonlinear stability of the scheme and to enhance the efficiency.