Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations
Applied Numerical Mathematics - Special issue on time integration
Linearly implicit Runge—Kutta methods for advection—reaction—diffusion equations
Applied Numerical Mathematics
Nodal high-order methods on unstructured grids
Journal of Computational Physics
Additive Runge-Kutta schemes for convection-diffusion-reaction equations
Applied Numerical Mathematics
Journal of Computational Physics
Principles of Computational Fluid Dynamics
Principles of Computational Fluid Dynamics
Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes
Journal of Computational Physics
Journal of Scientific Computing
International Journal of Numerical Modelling: Electronic Networks, Devices and Fields
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Numerical Integration of Damped Maxwell Equations
SIAM Journal on Scientific Computing
Energy Conserving Explicit Local Time Stepping for Second-Order Wave Equations
SIAM Journal on Scientific Computing
Locally implicit discontinuous Galerkin method for time domain electromagnetics
Journal of Computational Physics
Explicit local time-stepping methods for Maxwell's equations
Journal of Computational and Applied Mathematics
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An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit---explicit approaches in presence of local refinements.