TY - JOUR

T1 - Analysis of the convergence properties for a non-linear implicit Equilibrium Flux Method using Quasi Newton–Raphson and BiCGStab techniques

AU - Smith, Matthew R.

AU - Lin, Yi Hsin

N1 - Funding Information:
The authors would like to acknowledge support from the R.O.C. (Taiwan) Ministry of Science and Technology (MOST) under grant MOST 103-2221-E-006 -097 -MY2 for this research, which is greatly appreciated. The authors would also like to acknowledge support from Taiwan’s National Centre for High-performance Computing (NCHC) for use of their supercomputing facilities.
Publisher Copyright:
© 2016 Elsevier Ltd

PY - 2016/10/1

Y1 - 2016/10/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.camwa.2016.05.019

DO - 10.1016/j.camwa.2016.05.019

M3 - Article

AN - SCOPUS:84992529784

VL - 72

SP - 2008

EP - 2019

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 8

ER -