A study of numerical methods for hyperbolic conservation laws with stiff source terms
Journal of Computational Physics
Multidimensional upwind methods for hyperbolic conservation laws
Journal of Computational Physics
SIAM Journal on Scientific Computing
Two-dimensional front tracking based on high resolution wave propagation methods
Journal of Computational Physics
SIAM Journal on Scientific Computing
A pseudo-non-time-splitting method in air quality modeling
Journal of Computational Physics
An implicit-explicit approach for atmospheric transport-chemistry problems
Applied Numerical Mathematics
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
A semi-implicit numerical scheme for reacting flow: I. stiff chemistry
Journal of Computational Physics
Proceedings of the on Numerical methods for differential equations
An analysis of operator splitting techniques in the stiff case
Journal of Computational Physics
Journal of Computational Physics
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Additive Runge-Kutta schemes for convection-diffusion-reaction equations
Applied Numerical Mathematics
Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics
Journal of Computational Physics
Semi-implicit projection methods for incompressible flow based on spectral deferred corrections
Applied Numerical Mathematics - Special issue: Workshop on innovative time integrators for PDEs
Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics
Journal of Computational Physics
A fourth-order accurate local refinement method for Poisson's equation
Journal of Computational Physics
Accelerating the convergence of spectral deferred correction methods
Journal of Computational Physics
An integral equation method for epitaxial step-flow growth simulations
Journal of Computational Physics
Arbitrary order Krylov deferred correction methods for differential algebraic equations
Journal of Computational Physics
Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics
Journal of Computational Physics
Error estimates for deferred correction methods in time
Applied Numerical Mathematics
A fourth-order auxiliary variable projection method for zero-Mach number gas dynamics
Journal of Computational Physics
On the choice of correctors for semi-implicit Picard deferred correction methods
Applied Numerical Mathematics
Applied Numerical Mathematics
Applied Numerical Mathematics
Journal of Computational and Applied Mathematics
Semi-implicit Krylov deferred correction methods for ordinary differential equations
AMATH'09 Proceedings of the 15th american conference on Applied mathematics
L2-stability analysis of novel ETD scheme for Kuramoto-Sivashinsky equations
Journal of Computational and Applied Mathematics
Solvability of concatenated Runge-Kutta equations for second-order nonlinear PDEs
Journal of Computational and Applied Mathematics
Accelerating moderately stiff chemical kinetics in reactive-flow simulations using GPUs
Journal of Computational Physics
Hi-index | 31.49 |
Models for reacting flow are typically based on advection-diffusion-reaction (A-D-R) partial differential equations. Many practical cases correspond to situations where the relevant time scales associated with each of the three sub-processes can be widely different, leading to disparate time-step requirements for robust and accurate time-integration. In particular, interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection. The numerical strategy introduced in this paper is a general procedure to account for this time-scale disparity. The proposed methods are high-order multi-implicit generalizations of spectral deferred correction methods (MISDC methods), constructed for the temporal integration of A-D-R equations. Spectral deferred correction methods compute a high-order approximation to the solution of a differential equation by using a simple, low-order numerical method to solve a series of correction equations, each of which increases the order of accuracy of the approximation. The key feature of MISDC methods is their flexibility in handling several sub-processes implicitly but independently, while avoiding the splitting errors present in traditional operator-splitting methods and also allowing for different time steps for each process. The stability, accuracy, and efficiency of MISDC methods are first analyzed using a linear model problem and the results are compared to semi-implicit spectral deferred correction methods. Furthermore, numerical tests on simplified reacting flows demonstrate the expected convergence rates for MISDC methods of orders three, four, and five. The gain in efficiency by independently controlling the sub-process time steps is illustrated for nonlinear problems, where reaction and diffusion are much stiffer than advection. Although the paper focuses on this specific time-scales ordering, the generalization to any ordering combination is straightforward.