Stabilized Finite Element Schemes for Hyperbolic Conservation Laws

Abstract:  An accurate solution of hyperbolic conservation laws is important in many scenario for example, high speed flows governed by Euler equations of gas dynamics, shallow river flows, open channel flows governed by shallow water equations, astrophysical flows governed by Magnetohydrodynamic equations etc. These equations describe the propagation of linear and nonlinear waves in space and time. Due to nonlinear nature of convection term such equations admits discontinuous solution hence, an accurate numerical methods are required to solve these equations. Finite element method is a versatile method to solve the partial differential equations. For the solution of hyperbolic conservation laws special class of finite element methods also called as stabilized finite element methods are used. In this talk a novel stabilized finite element schemes developed for hyperbolic conservation laws will be presented. Due to stability reasons, time stepping of the numerical scheme designed for hyperbolic conservation laws is restricted by the CFL criteria. In this work spectral stability analysis is also done which gives an implicit expression for stable time step. Moreover, error analysis gives the convergence rate for the proposed schemes. Various test cases are solved which validates the numerical results and shows the robustness, efficiency and accuracy of the proposed schemes.