The semi-implicit Euler discretization method is studied for abstract evolution equations in a Hilbert space H, like _ u = f(t; u; u) ; t 2 (0; T ] ; u(0) = u0, where f(t; ; v) is one-sided Lipschitz and R(I hf(t; ; v)) = H for h > 0 su ciently small, and f(t; u; ) is Lipschitz-continuous. Extension to Banach spaces is then pointed out. Ordinary and partial , di erential and integro-di erential...

We explain an interface for the implementation of rate-independent elastoplasticity which separates the pointwise evaluation of the elastoplastic material law and the global solution of the momentum balance equation. The elastoplastic problem is discretized in time by the implicit Euler method and every time step is solved with a Newton iteration. For the discretization in space the material pa...

In population balance equations, the distribution of the entities depends not only on space and time but also on their own properties referred to as internal coordinates. The operator splitting method is used to transform the whole time-dependent problem into two unsteady subproblems of a smaller complexity. The first subproblem is a timedependent convection-diffusion problem while the second o...

We consider the numerical solution of hyperbolic systems of conservation laws with relaxation using a shock capturing nite diierence scheme on a xed, uniform spatial grid. We conjecture that certain a priori criteria insure that the numerical method does not produce spurious solutions as the relaxation time vanishes. One criterion is that the limits of vanishing relaxation time and vanishing vi...

We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order erro...

The classical two-phase Stefan problem in level set formulation is considered. The implementation of a solver on triangular grids is described. Extended finite elements (X-FEM) in space and an implicit Euler method in time are used to approximate the temperature. For the level set equation, a discontinuous Galerkin (DG) and a strong stability preserving (SSP) RungeKutta scheme are employed. Pol...

We propose an efficient and spectrally accurate numerical method for computing vortex lattice structures in fast rotating Bose-Einstein condensates (BECs) with strongly repulsive interactions. The key ingredients of the method is to discretize the normalized gradient flow under rotational frame by Fourier spectral method in space and by backward Euler method in time. Different vortex lattice st...

In this paper, an adaptive approach based on a time discontinuous Galerkin method (dG) is proposed. For large time steps and in a linear non-autonomous case, it reduces to the asymptotically second-order accurate Lobatto IIIC type implicit Runge-Kutta method, which is equal to the Padé-(0,2) approximation of the exponential function. In the asymptotic range it will result in the asymptotically ...

Asymptotic linear stability is studied for stochastic differential equations (SDEs) that incorporate Poisson-driven jumps and their numerical simulation using theta-method discretisations. The property is shown to have a simple explicit characterisation for the SDE, whereas for the discretisation a condition is found that is amenable to numerical evaluation. This allows us to evaluate the asymp...

In this work we present a framework for the convergence analysis in a measure differential inclusion sense of a class of time-stepping schemes for multibody dynamics with contacts, joints, and friction. This class of methods solves one linear complementarity problem per step and contains the semi-implicit Euler method, as well as trapezoidallike methods for which second-order convergence was re...