Abstract
Runge–Kutta time integration is used to reach the steady state solution of discretized partial differential equations. Continuous and discrete parameters in the method are adapted to the particular problem by minimizing the residual in each step, if this is possible, or the work to reach convergence. Algorithms for parameter optimization are devised and analyzed. Solutions of the linearized Euler equations and the nonlinear Euler and Navier–Stokes equations for compressible flow illustrate the methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Brown, P. N., and Saad, Y. (1990). Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11(3), 450–481.
Chiu, C., and Kopriva, D. A. (1992). An optimal Runge–Kutta method for steady-state solutions of hyperbolic systems. SIAM J. Numer. Anal. 29(2), 425–438.
Courant, R., Friedrichs, K., and Lewy, H. (1967). On the partial difference equations of mathematical physics. IBM J. 11, 215–234.
Golub, G. H., and Kannan, R. (1986). Convergence of a two-stage Richardson process for nonlinear equations. BIT 26(2), 209–216.
Greenbaum, A. (1997). Iteration Methods for Solving Linear Systems, SIAM, Philadelphia, PA.
Gustafsson, B., Kreiss, H.-O., and Oliger, J. (1996). Time Dependent Problems and Difference Methods, Wiley, New York.
Hackbusch, W. (1985). Multigrid Methods and Applications, Springer-Verlag, Berlin/Heidelberg.
Hairer, E., and Wanner, G. (1996). Solving Ordinary Differential Equations II—Stiff and Differential-Algebraic Problems, 2nd ed., Springer-Verlag, Berlin.
Hirsch, C. (1990). Numerical Computation of Internal and External Flows, Vols. 1 and 2, Wiley, New York.
Hörnell, K. (1999). Runge–Kutta Time Step Selection for Flow Problems, Ph.D. thesis, Dept. of Scientific Computing, Uppsala University, Uppsala, Sweden.
Jameson, A. (1983). Solution of the Euler equations for two-dimensional transonic flow by a multigrid method. Appl. Math. Comput. 13, 327–355.
Jameson, A., Schmidt, W., and Turkel, E. (1981). Numerical solutions of the Euler equations by the finite volume method using Runge–Kutta time-stepping schemes. AIAA paper 81-1259.
van Leer, B., Lee, W.-T., Roe, P. L., Powell, K. G., and Tai, C.-H. (1992). Design of optimally smoothing multistage schemes for the Euler equations. Comm. Appl. Numer. Methods 8, 761–769.
Lötstedt, P. (1994). Improved convergence to the steady state of the Euler equations by enhanced wave propagation. J. Comput. Phys. 114(1), 34–44.
Luenberger, D. G. (1973). Linear and Nonlinear Programming, Addison–Wesley, Reading, Mass.
Lynn, J. F., and van Leer, B. (1993). Multi-stage schemes for the Euler and Navier–Stokes equations with optimal smoothing. AIAA paper 93-3355.
Martinelli, L., and Jameson, A. (1988). Validation of a multigrid method for the Reynolds averaged equations. AIAA paper 88-0414.
Mavriplis, D. J., and Jameson, A. (1990). Multigrid solution of the Navier–Stokes equations on triangular meshes. AIAA J. 28(8), 1415–1425.
Müller, B. (1990). Linear stability condition for explicit Runge–Kutta methods to solve the compressible Navier–Stokes equations. Math. Meth. Appl. Sci. 12, 139–151.
Nachtigal, N. M., Reichel, L., and Trefethen, L. N. (1992). A hybrid GMRES algorithm for nonsymmetric linear systems. SIAM J. Matrix Anal. Appl. 13(3), 796–825.
Opfer, G. (1984). Richardson's iteration for nonsymmetric matrices. Lin. Alg. Appl. 58, 343–361.
Otto, K. (1996). Analysis of preconditioners for hyperbolic partial differential equations. SIAM J. Numer. Anal. 33(6), 2131–2165.
Saylor, P. E., and Smolarski, D. C. (1991). Implementation of an adaptive algorithm for Richardson's method. Lin. Alg. Appl. 154/156, 515–646.
Swanson, R. C., and Turkel, E. (1985). A multistage time-stepping scheme for the Navier–Stokes equations. AIAA paper 85-0035.
Swanson, R. C., and Radespiel, R. (1991). Cell centered and cell vertex multigrid schemes for the Navier–Stokes equations. AIAA J. 29(5), 697–703.
Reddy, S. C., and Trefethen, L. N. (1992). Stability of the method of lines. Numer. Math. 62, 235–267.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hörnell, K., Lötstedt, P. Adaptive Iteration to Steady State of Flow Problems. Journal of Scientific Computing 20, 331–354 (2004). https://doi.org/10.1023/B:JOMP.0000025928.28208.32
Issue Date:
DOI: https://doi.org/10.1023/B:JOMP.0000025928.28208.32