Numerical Computations with Fundamental Solutions

Abstract: Two solution strategies for large, sparse, and structured algebraic systems of equations are considered. The first strategy is to construct efficient preconditioners for iterative solvers. The second is to reduce the sparse algebraic system to a smaller, dense system of equations, which are called the boundary summation equations.The proposed preconditioners perform well when applied to equations that are discretizations of linear first order partial differential equations. Analysis shows that also very simple iterative methods converge in a number of iterations that is independent of the number of unknowns, if our preconditioners are applied to certain scalar model problems. Numerical experiments indicate that this property holds also for more complicated cases, and a flow problem modeled by the nonlinear Euler equations is treated successfully.The reduction process is applicable to a large class of difference equations. There is no approximation involved in the reduction, so the solution of the original algebraic equations is determined exactly if the reduced system is solved exactly. The reduced system is well suited for iterative solution, especially if the original system of equations is a discretization of a first order differential equation. The technique is used for several problems, ranging from scalar model problems to a semi-implicit discretization of the compressible Navier-Stokes equations.Both strategies use the concept of fundamental solutions, either of differential or difference operators. An algorithm for computing fundamental solutions of difference operators is also presented.