We introduce the concept of fast wavelet-Taylor Galerkin methods for the numerical solution of partial differential equations. In wavelet-Taylor Galerkin method discretization in time is performed before the wavelet based spatial approximation by introducing accurate generalizations of the standard Euler, and leap-frog time-stepping scheme with the help of Taylor series expansions in the time s...

Semi-discrete Galerkin formulations of transient wave equations, either with conforming or discontinuous Galerkin finite element discretizations, typically lead to large systems of ordinary differential equations. When explicit time integration is used, the time-step is constrained by the smallest elements in the mesh for numerical stability, possibly a high price to pay. To overcome that overl...

The application of standard multigrid methods for the solution of the Navier-Stokes equations in complicated domains causes problems in two ways. First, coarsening is not possible to full extent since the geometry must be resolved by the coarsest grid used, and second, for semiimplicit time stepping schemes, robustness of the convergence rates is usually not obtained for the arising convection-...

The numerical approximations to the Allen–Cahn type diffuse interface models are studied, with a particular focus on their performance in the sharp interface limit and the effectiveness of high order discretization schemes. Different spatial discretizations of an energy functional in the diffuse interface framework are compared first. Discretizations of the time-dependent equation using various...

Water movement in an unsaturated porous medium (soil) can be expressed by Richards equation with the mass conservation law and Darcy-Buckingham's law. This three different forms as pressure head-based, moisture content based mixed from. In this study, we solve one dimensional Equation form numerically using finite difference method various time-stepping schemes: Forward Euler, Backward Crank-Ni...

The immersed boundary method is known to exhibit a high degree of numerical sti ness, which is associated with the interaction of immersed elastic bres with the surrounding uid. We perform a linear analysis of the underlying equations of motion for immersed bres, and identify a discrete set of bre modes which are associated solely with the presence of the bre. These results are a generalisation...

We are concerned with the solution of time-dependent nonlinear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge-Kutta-type time stepping (discretisation in time). The introduced nonlinear blending procedure allows us to retain the explicit character of the time stepping ...

The aim of this paper is to build and validate some explicit high-order schemes, both in space and time, for simulating the dynamics of systems of nonlinear Schrödinger/Gross-Pitaevskii equations. The method is based on the combination of high-order IMplicit-EXplicit (IMEX) schemes in time and Fourier pseudo-spectral approximations in space. The resulting IMEXSP schemes are highly accurate, eff...

Numerical schemes for Einstein’s vacuum equation are developed. Einstein’s equation in harmonic gauge is second order symmetric hyperbolic. It is discretized in four-dimensional spacetime by Finite Differences, Finite Elements, and Interior Penalty Discontinuous Galerkin methods, the latter related to Regge calculus. The schemes are split into space and time and new time-stepping schemes for wa...

The application of nonlinear schemes like dual time stepping as preconditioners in matrix-free Newton– Krylov-solvers is considered and analyzed, with a special emphasis on unsteady viscous flows. We provide a novel formulation of the left preconditioned operator that says it is in fact linear in the matrix-free sense, but changes the Newton scheme. This allows to get some insight in the conver...