Analysis of the convergence properties for a non-linear implicit Equilibrium Flux Method using Quasi Newton-Raphson and BiCGStab techniques

In this work, an investigation is carried out into the implicit formulation of the Equilibrium Flux Method (EFM) applied to the numerical solution of the Euler Equations for an ideal inviscid gas. The discretization employs a non-linear Finite Volume Method (FVM) approach which requires the solution to a system of non-linear equations, which is solved using various modifications to the Quasi-Newton-Raphson method. The core of the analysis presented here lies in the investigation of the convergence properties of the BiConjugate Stabilized (BiCGStab) method used to solve the JΔx=R where R is our vector of residuals computed directly using the Euler Equations, Δx is the increment to our estimated solution x for our system and J is the Jacobian of the residual functions. The increment to our solution Δx is modified to ensure the solution remains bound and finite. Results are shown for multiple one dimensional and two dimensional benchmark problems, with convergence properties of both the Newton-Raphson and BiCGStab procedures shown. Several conclusions may be drawn from the results shown: (a) increasing CFL numbers result in increasing dispersion errors, (b) higher order treatment of temporal derivatives results in lower condition numbers, resulting in fewer BiCGstab iterations per Newton-Raphson iteration, (c) higher order treatment of spatial derivatives results in higher condition numbers, hence requiring additional BiCGstab iterations, and (d) preconditioning using a simple Jacobi preconditioner significantly reduces the number of BiCGstab iterations required to obtain a solution.