High Order Residual Distribution Methods For Steady State Problems

Many physics problems are related to hyperbolic conservation laws,such as,fluid dynamics,gas dynamics,shallow water and combustion,etc.Unfortunately,most of partial differential equations are unable to get the exact solutions,so we only can use numerical methods to find their approximated solutions.Especially,the solutions of nonlinear hyperbolic conservation law problems may produce intermittent,regardless of the smooth initial condition,for instance,shock and contact discontinuity and so on,which is a quite challenge for many researchers to develop numerical methods that can efficiently capture shocks in a sharp,non-oscillatory fashion and also maintain high order accuracy in smooth regions,and to do numerical simulations.However,the quality of the solutions of many current schemes is still questionable.In fact they have been based on one dimensional scalar problems,then extended to Multi-Dimension systems,but their construction relies on“1D ideas".Another difficult problem is the sensitivity to the mesh.Therefore,we propose Residual Distribution(RD)methods which have no request of regularity of mesh and can easily achieve any high order accuracy and it is based on a Multi-resolution WENO(Weighted Essentially Non-Oscillatory)integration.In addition,a Hermite WENO(HWENO)integration is also combined with RD method.In this dissertation,we propose several high order RD conservative finite difference methods for solving steady state conservation laws.This thesis is organized as follows:In Chapter 1,a brief history of numerical methods for solving conservation laws problems,then the existing works and research motivation for RD schemes.In Chapter 2,we propose fourth order and sixth order RD conservative finite difference schemes for solving steady state conservation laws.A new multi-resolution WENO integration is used to compute the numerical fluxes and source term based on the point values of the solution,and an upwind residual distribution scheme is adapted to obtain steady state solutions.Given the structured meshes,a dimension-by-dimension fashion can be used for two-dimensional problems.Extensive numerical examples in both scalar and systems in one and two dimensions demonstrate the efficiency,high order accuracy and the capability of capturing shocks of the proposed method.In Chapter 3,we propose fourth order and sixth order RD conservative finite difference scheme for solving steady state conservation laws.A multi-resolution WENO integration is still used to compute the numerical fluxes and source term based on the point values of the solution.Instead of the upwind residual distribution scheme proposed in Chapter 2,we use a SUPG(Streamline Upwind Petrov Galerking)type residual distribution scheme,which is parameter-free and can easily extend to Multi-dimension systems.Many numerical examples in both scalar and systems in one and two dimensions demonstrate the efficiency,high order accuracy and the capability of capturing shocks of the proposed method.In Chapter 4,we propose a sixth order RD conservation finite difference scheme for solving steady state conservation laws.A Hermite WENO integration is used to compute the source term based on the point values of the solution and its derivative,which is more compact than WENO scheme,and then use the same upwind RD scheme and SUPG-like RD scheme to distribute the residuals and then get steady state solutions,respectively.Some numerical examples in both scalar and systems in one dimension demonstrate the efficiency,high order accuracy and the capability of capturing shocks of proposed method.