Design of Algorithms for a Dispersive Hyperbolic Problem b9'- Y 8 7i5 Philip L. Roe* and Mohit Arorat The University of Michigan Ann Arbor, Michigan Abstract In order to develop numerical schemes for stiff prob- lems, we have studied a model of relaxing heat flow. To isolate those errors unavoidably associated with dis- cretization, a method of characteristics is developed, containing three free parameters depending on the stiff- ness ratio. It is shown that such 'decoupled' schemes do not take into account the interaction between the wave families and hence result in incorrect wavespeeds. We also demonstrate that schemes can differ by up to two orders of magnitude in their rms errors even while maintaining second order accuracy. Next, we develop 'coupled' schemes which account for the interactions, and here we obtain two additional free parameters. We present numerical results for several decoupled and cou- pled schemes. 1 Introduction In the real-life problem of attempting to solve the re- active flow equations, we often have a so-called 'stiff' problem. The flow equations are constrained to a maximum time-step by the CFL condition. Unfortu- nately, such large time-steps may be unacceptable for the chemical reactions. We are usually left with two choices - split the chemistry from the flow for each time step, or use the time-step dictated by the chemistry to solve the entire problem. The former reduces the cred- ibility of the results, while the latter is prohibitively expensive computationally for obvious reasons. It was this impasse that motivated the present work. Instead of tackling the full set of reactive flow equations right off, a simple model was derived. We expected valuable insight into the real problem since its disper- sive wave characteristics are akin to those of a reactive flow with the added advantages of minimal computa- tional cost and greatly simplified analysis. We started with the conservation of energy in a uniform conduct- ing rod with heat flow. Instead of using Fourier's Law 'Professor, Aerospace Engineering 'Doctoral Pre-candidate, Aerospace Engineering of Heat Conduction, which would lead to the parabolic heat equation, we use a simple model of heat conduc- tion that has a relaxation time T, in which information propagates at a finite speed. The result is what we call the Hyperbolic Heat Equations. T = 0, oo give us the extrema: in the first case, we have Fourier's Law with an infinite propagation speed, and in the second case, we have no propagation of information. These relations are examined using dispersion analysis, and a transformation to characteristic coordinates gives us the characteristic equations and jump relations. We now sought to isolate the effect of stiffness on the quality of the numerical solution. We decided to use the Method of Characteristics, which is exact for linear problems without source terms, so that numerical diffi- culties would arise only due to the presence of the stiff source terms. A straightforward discretization results in the appearance of a 'stiffness factor' k in a natural manner. Due to this factor, we expected problems for k > 1, as some terms would change sign in this sim- ple discretization. Another scheme we tried out was a simple one used in practice - symmetric operator split- ting. Here, no terms change sign, but only the frozen wavespeeds are allowed for. Our initial hypothesis was that since the source term prevents the characteristic equations from being inte- grated exactly, success, if achievable, would come from finding the best approximate quadrature. To this end, we rewrote the Method of Characteristics with three free parameters, depending on k alone, and sought to determine these by various heuristic arguments, mak- ing numerical trials of the resulting schemes. Our plan was to identify the successful heuristics, and then apply them to more complex sets of equations. Constraints on the parameters were derived from discrete dispersion relationships, discrete eigenvectors, local truncation er- rors, a modified conservation as well as a decoupling condition. These constraints are found to be consistent for small k but to suggest contradictory design criteria for large k. We ran tests on the various schemes varying k. Remarkably, our results show that the errors from various equally plausible characteristic schemes can eas- 2opyright 01991 by the American Institute of Aeronautics 95 ~nd Astronautics, Inc. All rights reserved.