A preconditioning mass matrix to accelerate the convergence to the steady Euler solutions using explicit schemes. (English) Zbl 0772.76054

The authors present, with analytical and numerical results, how the use of a preconditioning mass matrix (PMM) accelerates the convergence in a prescribed range of Mach numbers. It has been also claimed that PMM can be applied to any FEM/FVM that uses an explicit time marching scheme to find the steady state. The rate of convergence to the steady state is also studied and the results for the one- and two-dimensional cases are presented. Finally, it is concluded that the numerical scheme presented is very efficient in terms of computing time, simplicity in programming and is also as robust as the original scheme.


76M20 Finite difference methods applied to problems in fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
Full Text: DOI Link


[1] and , ’Comparison of finite flux vector splittings for the Euler equations’, AIAA J., 1453–1460 (1986).
[2] Steger, J. Comp. Phys. 40 pp 263– (1981)
[3] Trefethen, SIAM Rev. 24 pp 113– (1982)
[4] and , Fourier Analysis of Numerical Approximations of Hyperbolic Equations, SIAM, Philadelphia, 24, 1982.
[5] Harten, J. Comp. Phys. 49 pp 151– (1983)
[6] Tadmor, J. Math. Anal. Appl. 103 pp 428– (1984)
[7] Hughes, Compt. Methods Appl. Mech. Eng. 54 pp 223– (1986)
[8] and , ’Preconditioning mass matrices for hyperbolic systems’, GTM internal communication.
[9] Beam, J. Comp. Phys. 48 pp 200– (1982)
[10] ’Numerical boundary conditions’, Large-Scale Computations in Fluid Mechanics-Lectures in Applied Mathematics Vol. 22 -Part 1, American Mathematical Society, Providence, (1985).
[11] ’Stability of hyperbolic finite-difference models with one or two boundaries’, Computations in Fluid Mechanics-Lectures in Applied Mathematics Vol. 22 -Part 2, American Mathematical Society, Providence (1985).
[12] and , ’Stability analysis for the calculation of transonic flows with SUPG-type schemes’, GTM internal communication.
[13] and , ’Euler Computations of AGARD Working Group 07 Airfoil Test Cases’, AIAA Paper 85-0018, 1985.
[14] Engelman, Int. j. numer. methods fluids 2 pp 225– (1982)
[15] Baumann, Cuadernos Matematica Mecanica 3 (1990)
[16] and , ’Absorbing boundary conditions for the solution of transonic flows’, GTM internal communication.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.