Related papers: Structure preserving approximation of dissipative …
We consider evolutionary equations as introduced by R.\ Picard in 2009 and develop a general theory for approximation which can be seen as a theoretical foundation for numerical analysis for evolutionary equations. To demonstrate the…
A method for numerical approximation of a new class of fractional parabolic stochastic evolution equations is introduced and analysed. This class of equations has recently been proposed as a space-time extension of the SPDE-method in…
We develop an entropy-stable nodal discontinuous Galerkin (DG) scheme for the Euler equations with gravity, which is also well-balanced with respect to general equilibrium solutions, including both hydrostatic and moving equilibria. The…
We prove stability and convergence of a full discretization for a class of stochastic evolution equations with super-linearly growing operators appearing in the drift term. This is done using the recently developed tamed Euler method, which…
We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently…
We design a novel provably stable discontinuous Galerkin spectral element (DGSEM) approximation to solve systems of conservation laws on moving domains. To incorporate the motion of the domain, we use an arbitrary Lagrangian-Eulerian…
In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve…
We develop structure-preserving numerical methods for the compressible Euler equations, employing potential temperature as a prognostic variable. We construct three numerical fluxes designed to ensure the conservation of entropy and total…
We apply the Postprocessing Galerkin method to a recently introduced continuous data assimilation (downscaling) algorithm for obtaining a numerical approximation of the solution of the two-dimensional Navier-Stokes equations corresponding…
In this paper we introduce a procedure, based on the method of equivariant moving frames, for formulating continuous Galerkin finite element schemes that preserve the Lie point symmetries of initial value problems for ordinary differential…
Optimal Dirichlet boundary control for a fractional/normal evolution with a final observation is considered. The unique existence of the solution and the first-order optimality condition of the optimal control problem are derived. The…
This work concerns the analysis of the discontinuous Galerkin spectral element method (DGSEM) with implicit time stepping for the numerical approximation of nonlinear scalar conservation laws in multiple space dimensions. We consider either…
We consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We approximate their semiflow by an…
Numerical methods of approximate solution of the Cauchy problem for coupled systems of evolution equations are considered. Separating simpler subproblems for individual components of the solution achieves simplification of the problem at a…
We use a concept of weak asymptotic solution for homogeneous as well as non-homogeneous fractional advection dispersion type equations. Using Legendre scaling functions as basis, a numerical method based on Galerkin approximation is…
Wave propagation problems for heterogeneous media are known to have many applications in physics and engineering. Recently, there has been an increasing interest in stochastic effects due to the uncertainty, which may arise from impurities…
Energy methods for constructing time-stepping algorithms are of increased interest in application to nonlinear problems, since numerical stability can be inferred from the conservation of the system energy. Alternatively, symplectic…
The DG algorithm is a powerful method for solving pdes, especially for evolution equations in conservation form. Since the algorithm involves integration over volume elements, it is not immediately obvious that it will generalize easily to…
We propose a fully discrete variational scheme for nonlinear evolution equations with gradient flow structure on the space of finite Radon measures on an interval with respect to a generalized version of the Wasserstein distance with…
We propose and analyse numerical schemes for a system of quasilinear, degenerate evolution equations modelling biofilm growth as well as other processes such as flow through porous media and the spreading of wildfires. The first equation in…