We formulate collocation Runge–Kutta time-stepping schemes applied to linear parabolic evolution equations as space-time Petrov–Galerkin discretizations, and investigate their a priori stability for the parabolic space-time norms, that is the continuity constant of the discrete solution mapping. We focus on collocation based on A-stable Gauss–Legendre and L-stable right-Radau nodes, addressing ...

It is shown that for a broad class of equations that numerical solutions computed using the discontinuous Galerkin or the continuous Galerkin time stepping schemes of arbitrary order will inherit the compactness properties of the underlying equation. Convergence of numerical schemes for a phase field approximation of the flow of two fluids with surface tension is presented to illustrate these r...

In this paper, we propose a wavelet-Taylor–Galerkin method for solving the twodimensional Navier–Stokes equations. The discretization in time is performed before the spatial discretization by introducing second-order generalization of the standard time stepping schemes with the help of Taylor series expansion in time step. WaveletTaylor–Galerkin schemes taking advantage of the wavelet bases cap...

A review of a procedure for the simulation of time-dependent, inviscid and turbulent viscous, compressible flows involving geometries that change in time is presented. The adopted discretization technique employs unstructured meshes and both explicit and implicit time-stepping schemes. A dual time-stepping procedure and an ALE formulation enable flows involving moving boundary components to be ...

Abstract. This paper provides convergence analysis of regularized time-stepping methods for the differential variational inequality (DVI), which consists of a system of ordinary differential equations and a parametric variational inequality (PVI) as the constraint. The PVI often has multiple solutions at each step of a time-stepping method and it is hard to choose an appropriate solution for gu...

We develop upwind methods which use limited high resolution corrections in the spatial discretization and local time stepping for forward Euler and second order time discretizations. L∞ stability is proven for both time stepping schemes for problems in one space dimension. These methods are restricted by a local CFL condition rather than the traditional global CFL condition, allowing local time...

The continuous and discontinuous Galerkin time stepping methodologies are combined to develop approximations of second order time derivatives of arbitrary order. This eliminates the doubling of the number of variables that results when a second order problem is written as a first order system. Stability, convergence, and accuracy, of these schemes is established in the context of the wave equat...

The continuous and discontinuous Galerkin time stepping methodologies are combined to develop approximations of second order time derivatives of arbitrary order. This eliminates the doubling of the number of variables that results when a second order problem is written as a first order system. Stability, convergence, and accuracy, of these schemes is established in the context of the wave equat...

The majority of industrial computational fluid dynamics simulations are steady. Such simulations therefore require numerical methods which can rapidly converge a boundary value problem, typically either the Euler or Reynolds-averaged Navier–Stokes (RANS) equations. In this chapter we outline two major approaches: time-marching methods built on top of Runge–Kutta time-stepping schemes and Newton...