We analyse two practical aspects that arise in the numerical solution of HamiltonJacobi-Bellman (HJB) equations by a particular class of monotone approximation schemes known as semi-Lagrangian schemes. These schemes make use of a wide stencil to achieve convergence and result in discretization matrices that are less sparse and less local than those coming from standard finite difference schemes...

This work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented Monte Carlo Euler–Maruyama method, which achieve a prescribed precision. The main result is the derivation of new expansions for the time discretization error, with computable leading order term in a posteriori form, which are based on stochastic flows a...

We present a numerical comparison of some time{stepping schemes for the discretization and solution of the nonstationary incompressible Navier{Stokes equations. The spatial discretization is by nonconforming quadrilateral nite elements which satisfy the LBB{condition. The major focus is on the diierences in accuracy and eeciency between the Backward Euler-, Crank{Nicolson-or Fractional{step{{sc...

Electrical propagation (action potentials) in excitable tissue, such as nerve cells and cardiac myocytes, is described by a parabolic diffusion-reaction equation for the transmembrane potential (voltage) V (x, t), known as the cable equation. It is driven by a rapidly varying, highly nonlinear (and expensive to evaluate) ionic source term Iion(V, t) representing the total ionic current across t...

This paper presents the development of a 2D high-order solver with unstructured spectral difference method for unsteady incompressible Navier-Stokes equations. Timemarching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditiona...

A class of finite volume methods based on standard high resolution schemes, but which allows spatially varying time steps, is described and analyzed. A maximum principle and the TVD property are verified for general advective flux, extending the previous theoretical work on local time stepping methods. Moreover, an entropy condition is verified which, with sufficient limiting, guarantees conver...

Abstract. We present a new hp-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements, giving rise to prismatic space-time elements. A key feature of the proposed method is the use of space-time elemental polynomial bases of total degree...

We present a way of constructing multi-time-step monolithic coupling methods for elastodynamics. The governing equations for constrained multiple subdomains are written in dual Schur form and enforce the continuity of velocities at system time levels. The resulting equations will be in the form of differential-algebraic equations. To crystallize the ideas we shall employ Newmark family of time-...

Flow movement in unsaturated soil can be expressed by a partial differential equation, named Richards equation. The objective of this study is the finding of an appropriate implicit numerical solution for head based Richards equation. Some of the well known finite difference schemes (fully implicit, Crank Nicolson and Runge-Kutta) have been utilized in this study. In addition, the effects of di...

Abstract Numerical glacier and ice-sheet models compute evolving ice geometry velocity fields using various stress-balance approximations boundary conditions. At high spatial resolution, with horizontal mesh/grid resolutions of a few kilometers or smaller, these usually require time steps shorter than climate-coupling scales because they update thickness after each solution. High-resolution per...