Stability Analysis Of The Inverse Lax-Wendroff Boundary Treatment For High Order Finite Difference Schemes

When a high order finite difference scheme with wide stencil is used to solve par-tial differential equations,the inner scheme cannot be used near the boundary.We need to construct appropriate numerical boundary conditions to maintain accuracy and stability.This dissertation talks about stability analysis of the inverse Lax-Wendroff(ILW)boundary treatment and the simplified inverse Lax-Wendroff(SILW)boundary treatment for high order finite difference schemes for conservation laws and diffusion equations,which are initial-boundary value problems.Stability analysis is performed by the GKS(Gustafsson,Kreiss and Sundstrom)analysis and the eigenvalue spectrum visualization method.It mainly contains two parts.In the first part,we consider the case of high order upwind-biased finite difference schemes for one-dimensional hyperbolic conservation laws.There exists two difficul-ties when imposing numerical boundary conditions.Firstly,the points used in these schemes which lie outside the computational domain,namely "ghost points",should be evaluated properly.Secondly,the grid points may not coincide with the physical bound-ary exactly.These will cause trouble in constructing appropriate numerical boundary conditions.The ILW procedure and SILW procedure can overcome these difficulties and maintain stability and accuracy.The basic idea of the ILW procedure is to use Taylor expansion at the boundary point and then repeatedly use the PDE and its time derivatives to convert spatial derivatives to time derivatives,in order to obtain accurate values at the relevant ghost points.But if the equation is complex and for high order spatial derivatives,the procedure will be algebraically very complicated.A simplified version of ILW procedure,which is referred as simplified inverse Lax-Wendroff(SIL-W)procedure is used to save in algorithm complexity and computational cost.SILW procedure also uses Taylor expansion at the boundary point to evaluate the ghost points but the spatial derivatives at the boundary point are obtained in two ways:(i)spatial derivatives can be obtained through the ILW procedure(ii)derivatives can be obtained through the classical Lagrangian extrapolation.After imposing numerical boundary conditions,stability analysis is performed by GKS analysis and eigenvalue spectrum visualization method to get proper values of the parameters.Numerical tests are per-formed to show stability results predicted by the analysis.In the second part,we consider to use the SILW treatment in the case of one-dimensional diffusion equations.We consider the two cases of Dirichlet boundary and Neumann boundary conditions.Unlike the hyperbolic conservation law,which can yield all the derivatives by using the equation and the given boundary condition,in the case of Dirichlet boundary conditions,only even order derivatives can be obtained through the ILW procedure and in the case of Neumann boundary conditions,only odd order derivatives can be obtained through the ILW procedure.For both boundary con-ditions,we construct different methods to get derivatives at the boundary point and then use Taylor expansion to evaluate the ghost points.GKS analysis and eigenvalue spec-trum visualization method are used to get appropriate values of the parameters to ensure stability.Numerical examples are given to demonstrate the results.