Multiresolution schemes for the reactive Euler equations

We present multiresolution (MR) schemes for the efficient numerical solution of the one-dimensional system of the reactive Euler equations, which has possibly stiff source terms. The original version of the method was developed by A. Harten (1995, Comm. Pure Appl. Math. 48(12), 1305) for homogeneous hyperbolic conservation laws. By computing the cell average MR-representation of the solution, we obtain much information about the solution's regularity. This description of smoothness is then used to reduce the number of direct flux computations as well as the expensive high-order ENO (essentially nonoscillatory) reconstruction both of which are now performed only near discontinuities. Thereby the numerical solution procedure becomes considerably more efficient. In the present case of the reactive Euler equations, the average efficiency factor measured by counting the number of actual Aux computations ranges from about 5 to 12. This is on the same order of, and in some cases comes reasonably chose to, actual speed-up factors obtained by code timings, which were between 3 to 5. The MR overhead rate was about 10% for the ENO and 36% for TVD schemes, respectively. The quality of the solution is shown to be the same as that of the finest grid. Detailed numerical and performance results are shown for up to fourth-order accuracy, for source terms ranging from moderate to extremely Stiff. (C) 1999 Academic Press.