A General Theory of Local Preconditioning and Its Application to the 2D Ideal MHD Equations

In this paper, we describe a general theory of local preconditioning which assigns a unique preconditioner for any set of two-dimensional hyperbolic partial differential equations. The preconditioner is optimal in the sense that it gives the lowest possible condition number (the ratio of maximum to minimum wave speeds). Taking the two-dimensional magnetohydrodynamic (MHD) equations as an example, we show how the preconditioner can be derived both approximately and exactly, and present some results including a numerical experiment to demonstrate its effectiveness. Introduction Local preconditioning is a technique to equalize as much as possible all the wave speeds of PDEs, so that the maximum local Courant number can be taken for all the waves (not just for the fastest one as in local time stepping) and all the error modes propagate at the same rate thereby accelerating the convergence toward a steady state. A way to alter the transient solutions is to take a particular combination of the residual components to drive each solution component. This is done by multiplying the residual by a locally evaluated matrix (preconditioning matrix). Several such matrices have already been found and used in practice for the Euler or Navier-Stokes, 9 but a systematic procedure to derive a preconditioner for an arbitrary set of PDEs was not available. This makes it difficult to extend this very useful technique to other complicated but important PDEs such as the magnetohydrodynamic(MHD) equations. In this paper, we report further development of the theory of local preconditioning previously presented at the 32nd AIAA Fluid Dynamics Conference 2002. We now give a complete account of the general theory for constructing a unique local preconditioning matrix for any set of first-order PDEs in two dimensions. Although the theory gives very simple preconditioning Research Fellow, Member AIAA Professor, Fellow AIAA Graduate Student Research Assistant Professor, Fellow AIAA Copyright c © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. matrices for some PDEs such as the Euler equations, it could yield in other cases rather complicated matrices not suitable for practical application. The ideal MHD equations, in particular, do lead us to such impractical results. Therefore, in this paper, we turn our attention to a numerical (and/or approximate) construction of the preconditioner. As to approximation, our focus is low-Mach-number flows where the condition number becomes singular, and also where the accuracy of compressible solvers may deteriorate as it happens to the Euler equations. Preconditioners are known to cure such problems. Inspection of the wave pattern altered by a low-Mach-number MHD preconditioner shows that the low-speed singularity has been completely removed and all the wave speeds have successfully been made equal, giving the condition number = 1. Plots of the condition number over a range of Mach number are shown for the 2D MHD equations to demonstrate the effectiveness of the preconditioners derived from the theory. Also, we show that numerical implementation is also possible. Formulas necessary for numerical implementation are given. The basic idea behind our construction of local preconditioning matrices is to decompose the PDE (based on its steady form) into a certain number of hyperbolic (advection) equations and/or a certain number of elliptic (Cauchy-Riemann) subsystems, and modify the wave speed of each subproblem independently to achieve as equal wave speeds as possible. In the next section, we describe the decomposition of the steady equations which gives the building blocks of the preconditioner. In Section 3, the form of the preconditioner is defined. In Section 4, the formulas for the acceleration factors are given, which completes the description of the construction of preconditioners. In Section 5, then, the MHD system is analyzed as an example. In Section 6 and 7, we present some useful formulas for implementing the preconditioner derived from the general theory. In Section 8, finally, we report numerical experiment for MHD nozzle flows to demonstrate the effectiveness of the preconditioner.