Asymptotic Preserving Spectral Deferred Correction Methods For Hyperbolic Systems With Relaxation

In this thesis,basing on spectral deferred correction(SDC)methods,we construct a new asymptotic preserving scheme for hyperbolic systems of conservation laws with stiff relaxation terms.The designing principle of the new scheme is to treat the relax-ation term implicitly,and the convection terms explicitly named semi-implicit spectral deferred correction(semi-implicit SDC)methods.Semi-implicit spectral deferred cor-rection methods can be constructed easily and systematically for any order of accuracy.The research in this thesis is mainly divided into the following parts.In the first part,we introduce two asymptotic preserving schemes,the plitting and implicit-explicit Runge-Kutta methods.We use Jin-Xin semilinear hyperbolic relax-ation system to introduce splitting scheme and show that it can be calculated explicitly.We introduce the implicit-explicit Runge-Kutta methods for hyperbolic systems of con-servation laws with stiff relaxation terms and give third and fifth order examples.At the same time,we give the accuracy of order condition and schemes.In the second part,we construct the general semi-implicit spectral deferred cor-rection schemes and give the second and third order of semi-implicit SDC schemes.Furthermore,we deal with initial problem.In the third part,we introduce finite volume and finite difference methods,which are obtained by weighted essentially non-oscillatory(WEND)reconstruction and WENO-Z reconstruction in spatial discretization to achieve high accuracy.In the last part,we use broadwell equation,the linear wave of the relaxation sys-tem for the 1-D euler equations,the linear wave of the relaxation system for the 2-D euler equations and vortex evolution equation to test the accuracy of semi-implicit SDC schemes.Then we use shallow water equation,traffic flow equation,the continuum equations of Euler type for a granular gas,the Shock tube problem of the relaxation system for the 1-D Euler equations,the double mach reflection and a mach 3 wind tunnel with a step problem of the relaxation system for the 2-D Euler equations to test semi-implicit SDC schemes work well without spurious oscillations near discontinuities for piecewise smooth functions.