This study introduces new strategies for global error estimation in time-stepping algorithms. The new methods propagate the defect along with the numerical solution much like the Zadunaisky procedure; however, the proposed approach allows for overlapped internal computations and, therefore, represents a generalization of the classical numerical schemes for solving differential equations with gl...

Variational formulations of time-dependent PDEs in space and time yield (d + 1)-dimensional problems to be solved numerically. This increases the number unknowns as well storage amount. On other hand, this approach enables adaptivity model reduction w.r.t. both type variables. In paper, we show that matrix oriented techniques can significantly reduce computational timings for solving arising li...

We propose second order accurate discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle for general nonlinear convection-diffusion equations on unstructured triangular meshes. Motivated by genuinely high order maximum-principle-satisfying DG schemes for hyperbolic conservation laws [14, 26], we prove that under suitable time step restriction for forward Euler time stepping...

Frequency conversion in second-order nonlinear materials is sensitive to the phase velocities of interacting optical waves. Accurate modeling of such problems with the finite-difference time-domain method requires extremely fine grid resolutions to minimize numerical dispersion errors. We propose an alternative approach based on a pseudospectral time-domain (PSTD) method for solving the nonline...

Abstract. In this paper we develop a two-grid convergence theory for the parallel-in-time scheme known as multigrid reduction in time (MGRIT), as it is implemented in the open-source XBraid package [25]. MGRIT is a scalable and multi-level approach to parallel-in-time simulations that non-intrusively uses existing time-stepping schemes, and that in a specific two-level setting is equivalent to ...