Related papers: Multi-Adaptive Time-Integration
This paper develops a high order adaptive scheme for solving nonlinear Schrodinger equations. The solutions to such equations often exhibit solitary wave and local structures, which makes adaptivity essential in improving the simulation…
In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the…
Critical points of energy functionals, which are of broad interest, for instance, in physics and chemistry, in solid and quantum mechanics, in material science, or in general diffusion-reaction models arise as solutions to the associated…
We consider a sequence of elliptic partial differential equations (PDEs) with different but similar rapidly varying coefficients. Such sequences appear, for example, in splitting schemes for time-dependent problems (with one coefficient per…
Many applications from geosciences require simulations of seismic waves in porous media. Biot's theory of poroelasticity describes the coupling between solid and fluid phases and introduces a stiff source term, thereby increasing…
We present a family of multistep integrators based on the Adams-Bashforth methods. These schemes can be constructed for arbitrary convergence order with arbitrary step size variation. The step size can differ between different subdomains of…
Traditional time-domain discontinuous Galerkin (DG) methods result in large storage costs at high orders of approximation due to the storage of dense elemental matrices. In this work, we propose a weight-adjusted DG (WADG) methods for…
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…
Multiscale mathematical models have shown great promise in computational brain electrophysiology but are still hindered by high computational costs due to fast dynamics and complex brain geometries, requiring very fine spatio-temporal…
It is known in \cite{beccari} that the standard explicit Euler-type scheme (such as the exponential Euler and the linear-implicit Euler schemes) with a uniform timestep, though computationally efficient, may diverge for the stochastic…
We present a priori error estimates for a multirate time-stepping scheme for coupled differential equations. The discretization is based on Galerkin methods in time using two different time meshes for two parts of the problem. We aim at…
We study higher-order space-time variational discretisations for modeling complex processes in porous media that include fluid and structure interactions which are of fundamental importance in many engineering fields with applications in…
Explicit step-truncation tensor methods have recently proven successful in integrating initial value problems for high-dimensional partial differential equations (PDEs). However, the combination of non-linearity and stiffness may introduce…
Using existing, forward-in-time integration schemes, we demonstrate that it is possible to compute unstable, saddle-type fixed points of stiff systems of ODEs when the stable compenents are fast (i.e., rapidly damped) while the unstable…
Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is…
This paper studies the time-dependent test-function error in the characteristic Galerkin-type semi-Lagrangian discontinuous finite element (CSLDG) method caused by numerical integration errors of the characteristic ODE solver, and its…
The two-fluid plasma model has a wide range of timescales which must all be numerically resolved regardless of the timescale on which plasma dynamics occurs. The answer to solving numerically stiff systems is generally to utilize…
In this work, we present a new high order Discontinuous Galerkin time integration scheme for second-order (in time) differential systems that typically arise from the space discretization of the elastodynamics equation. By rewriting the…
The availability of data sets with large numbers of variables is rapidly increasing. The effective application of Bayesian variable selection methods for regression with these data sets has proved difficult since available Markov chain…
The goal of this paper is to create a fruitful bridge between the numerical methods for approximating partial differential equations (PDEs) in fluid dynamics and the (iterative) numerical methods for dealing with the resulting large linear…