High-order finite difference methods with subcell resolution for hyperbolic conservation laws with stiff reaction terms: preliminary results

The motivation for this research stems from the high-speed chemical reacting flows which have stiff reaction terms, where the chemical time scales are often much smaller than the fluid dynamics time scales. It is usually too expensive to resolve all the spatial/temporal scales if we are only interested in the main flow. On the other hand, insufficient spatial/temporal resolution will cause the speed of propagation of discontinuities to be incorrectly predicted for many numerical methods. This numerical phenomenon was first observed by Colella et al. (1986). Then LeVeque & Yee (1990) showed that a similar spurious propagation phenomenon can be observed even with scalar equations. Colella et al. (1986) and Majda & Roytburd (1990) have successfully applied the random choice method of Chorin (1976, 1977) for the solution of under-resolved detonation waves. However, it is difficult to eliminate completely the numerical viscosity in a shock-capturing scheme. Fractional step methods are commonly used for allowing an under-resolved mesh size with a shock-capturing method. Chang (1989, 1991) applied the subcell resolution method of Harten (1989) to the finite volume ENO method in the convection step, which is able to produce a zero viscosity shock profile in nonreacting flow. The time evolution is advanced along the characteristic line. Correct discontinuity speed was obtained in the one-dimensional scalar case. However, it is difficult to extend this approach to multi-dimensions and to system of equations because of the reliance on the exact time evolution via characteristics. Engquist & Sjögreen (1991) proposed a simple temperature extrapolation method based on finite difference ENO schemes with implicit Runge-Kutta time discretization, which uses a first-/second-order extrapolation of the temperatures from outside the shock profile. The method is easy to extend to multi-dimensions. Their method is not a fractional step method. It does not seem to work well when the spatial scales are under-resolved. Other first-/second-order methods that are based on the fractional step method have been proposed by Bao & Jin (2000, 2001) and Tosatto & Vigevano (2008). Our objective in this study is to develop a high-order finite difference method which can capture the correct detonation speed in an under-resolved mesh and will maintain highorder accuracy in the smooth part of the flow. The first step of the proposed fractional step method is the convection step which solves the homogeneous hyperbolic conservation law in which any high-resolution shock-capturing method can be used. The aim in this step is to produce a sharp wave front, but some dissipativity is allowed. The second