Numerical solution of under-resolved detonations

A new fractional-step method is proposed for the numerical solution of high speed reacting flows, where the chemical time scales are often much smaller than the fluid dynamical time scales. When the problem is stiff, because of insufficient spatial/temporal resolution, a well-known spurious numerical phenomenon occurs in standard finite volume schemes: the incorrect calculation of the speed of propagation of discontinuities. The new method is first illustrated considering a one-dimensional scalar hyperbolic advection/reaction equation with stiff source term, which may be considered as a model problem to under-resolved detonations. During the reaction step, the proposed scheme replaces the cell average representation with a two-value reconstruction, which allows us to locate the discontinuity position inside the cell during the computation of the source term. This results in the correct propagation of discontinuities even in the stiff case. The method is proved to be second-order accurate for smooth solutions of scalar equations and is applied successfully to the solution of the one-dimensional reactive Euler equations for Chapman-Jouguet detonations.

[1]  Randall J. LeVeque,et al.  A Modified Fractional Step Method for the Accurate Approximation of Detonation Waves , 2000, SIAM J. Sci. Comput..

[2]  Matania Ben-Artzi,et al.  The generalized Riemann problem for reactive flows , 1989 .

[3]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[4]  Andrew J. Majda,et al.  Theoretical and numerical structure for unstable one-dimensional detonations , 1991 .

[5]  Richard B. Pember,et al.  Numerical Methods for Hyperbolic Conservation Laws With Stiff Relaxation I. Spurious Solutions , 1993, SIAM J. Appl. Math..

[6]  P. Colella,et al.  Theoretical and numerical structure for reacting shock waves , 1986 .

[7]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .

[8]  E. F. Kaasschieter,et al.  Detonation capturing for stiff combustion chemistry , 1998 .

[9]  H. C. Yee,et al.  Numerical wave propagation in an advection equation with a nonlinear source term , 1992 .

[10]  Barna L. Bihari,et al.  Multiresolution Schemes for the Reactive Euler Equations , 1999 .

[11]  V. Ton,et al.  Improved Shock-Capturing Methods for Multicomponent and Reacting Flows , 1996 .

[12]  Alexandre J. Chorin,et al.  Random choice methods with applications to reacting gas flow , 1977 .

[13]  Shi Jin,et al.  The Random Projection Method for Stiff Detonation Capturing , 2001, SIAM J. Sci. Comput..

[14]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[15]  A. J. MAJDA,et al.  Numerical Study of the Mechanisms for Initiation of Reacting Shock Waves , 1990, SIAM J. Sci. Comput..

[16]  Rolf Jeltsch,et al.  Error estimators for the position of discontinuities in hyperbolic conservation laws with source terms which are solved using operator splitting , 1999 .

[17]  Weizhu Bao,et al.  The Random Projection Method for Hyperbolic Conservation Laws with Stiff Reaction Terms , 2000 .

[18]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[19]  Alexandre J. Chorin,et al.  Random choice solution of hyperbolic systems , 1976 .

[20]  D. Nguyen.  A Fully Conservative Ghost Fluid Method & Stiff Detonation Waves , 2002 .

[21]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[22]  Randall J. LeVeque,et al.  A study of numerical methods for hyperbolic conservation laws with stiff source terms , 1990 .