Date of this Version
Center for Turbulence Research Annual Research Briefs 2011
In simulating hyperbolic conservation laws in conjunction with an inhomogeneous stiff source term, if the solution is discontinuous, spurious numerical results may be produced due to the different time scales of the transport part and the source term. This numerical issue often arises in combustion and high-speed chemical reacting flows.
Our objective in this study is to extend this method to two-dimensional reactive Euler equations. 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 numerical dissipation is allowed. The second step is the reaction step where an ODE solver is applied with modified transition points. Here, by transition points, we refer to the smeared numerical solution in the shock region, which is due to the dissipativity of a shock-capturing scheme. Because the transition points in the convection step will result in large erroneous values of the source term if the source term is stiff, we first identify these points and then extrapolate them by a reconstructed polynomial using the idea of Harten’s subcell resolution method. Unlike Chang’s approach, we apply Harten’s subcell resolution in the reaction step. Thus our approach is flexible in allowing any shock-capturing scheme as the convection operator. In the reaction step, since the extrapolation is based on the high-order reconstruction, high-order accuracy can be achieved in space. The only drawback in our current approach is that the temporal accuracy will only be, at most, second-order due to the time splitting, which is common for most of the previous methods for stiff sources.We also remark that, in order to resolve the sharp reaction zone, sufficiently many grid points in this zone are still needed. The proposed method can capture the correct location and jump size of the reaction front, but it does not resolve the narrow reaction zone, as typically there is one or a few points in that zone.