The Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions (PNP-FBV) describe ion transport Faradaic reactions, and have applications in a number of fields. In this article, we develop an adaptive time-stepping scheme for the solution PNP-FBV system based on two methods: fully-implicit (VSBDF2) method, semi-implicit (VSSBDF2) method. We present simulations un...

In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, b...

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. Local time-stepping methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. Starting from classical Adams-Bashforth multi-step methods, local time-stepping methods of arbitr...

The goal of the paper is to bring to the attention of the computational community a long overlooked, very simple, acceleration method that impressively speeds up explicit time-stepping schemes, at essentially no extra cost. The authors explain the basis of the method, namely stabilization via wisely chosen inner steps (stages), justify it for linear problems, and spell out how simple it is to i...

Electrical propagation in excitable tissue, such as nerve fibers and heart muscle, is described by a parabolic PDE for the transmembrane voltage V (x, t), known as the cable equation,

Explicit local time-stepping methods are derived for the time dependent Maxwell equations in conducting and non-conducting media. By using smaller time steps precisely where smaller elements in the mesh are located, these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators. When combined with a finite element discretization in space with an essentially d...

Various recovery and stress-splitting schemes are investigated numerically for an Oldroyd-B model within the general framework of a time stepping fractional-staged finite element formulation, that of a Taylor-Galerkin/pressurecorrection method with consistent streamline upwinding. Smooth and non-smooth planar flows are cited and both creeping and inertial conditions are considered. Problems add...

• We derive simple and efficient formulas for Jacobian matrices of entropy stable schemes. These use automatic differentiation (AD) efficiently on functions with low input/output dimensions. The resulting provide up to 100x speedup when computing Jacobians high order are applied two-derivative implicit time-stepping Entropy schemes replicate an inequality at the semi-discrete level. rely algebr...