Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
On the contractivity of implicit-explicit linear multistep methods
Applied Numerical Mathematics
Additive Runge-Kutta schemes for convection-diffusion-reaction equations
Applied Numerical Mathematics
High-order multi-implicit spectral deferred correction methods for problems of reactive flow
Journal of Computational Physics
Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics
Journal of Computational Physics
Accelerating the convergence of spectral deferred correction methods
Journal of Computational Physics
Error estimates for deferred correction methods in time
Applied Numerical Mathematics
On the choice of correctors for semi-implicit Picard deferred correction methods
Applied Numerical Mathematics
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Many physical and biological systems involve the interactions of two or more processes with widely-differing characteristic time scales. Previously, high-order semi-implicit and multi-implicit formulations of the spectral deferred correction methods (denoted by SISDC and MISDC methods, respectively) have been proposed for solving partial differential equations arising in such model systems. These methods compute a temporally high-order approximation by means of a first-order numerical method, which solves a series of correction equations to increase the temporal order of accuracy of the approximation. MISDC methods also allow several fast-evolving processes to be handled implicitly but independently, allowing for different time steps for each process while avoiding the splitting errors present in traditional operator-splitting methods. In this study, we propose MISDC methods that use second- and third-order integration and splitting methods in the prediction steps, and we assess the efficiency of SISDC and MISDC methods that are based on those moderate-order integration methods. Numerical results indicate that SISDC methods using third-order prediction steps are the most efficient, but the efficiency of SISDC methods using first-order steps improves, particularly in higher spatial dimensions, when combined with a ''ladder approach'' that uses a less refined spatial discretization during the initial SDC iterations. Among the MISDC methods studied, the one with a third-order prediction step is the most efficient for a mildly-stiff problem, but the method with a first-order prediction step has the least splitting error and thus the highest efficiency for a stiff problem. Furthermore, a MISDC method using a second-order prediction step with Strang splitting generates approximations with large splitting errors, compared with methods that use a different operator-splitting approach that orders the integration of processes according to their relative stiffness.