In this paper, we analyze the stability of the fourth order Runge-Kutta method for integrating semi-discrete approximations of time-dependent partial differential equations. Our study focuses on linear problems and covers general semi-bounded spatial discretizations. A counter example is given to show that the classical four-stage fourth order Runge-Kutta method can not preserve the one-step st...

We devise a stabilized method to weakly enforce bound constraints in the discrete solution of advection-dominated diffusion problems. This combines nonlinear penalty formulation with discontinuous Galerkin-based residual minimization method. illustrate efficiency this scheme for both uniform and adaptive meshes through proper numerical examples.

We present a higher order accurate discontinuous Galerkin finite element method for the simulation of linear free-surface gravity waves. The method uses the classical Runge-Kutta method for the time discretization of the free-surface equations and the discontinuous Galerkin method for the space discretization. In order to circumvent numerical instabilities arising from an asymmetric mesh a stab...

We consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We study to what extent this property carries over to some piecewise linear finite element discretizations, namely the Standard Galerkin method, the Lumped Mas...

On the basis of a fully discrete trigonometric Galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order ‖uN − u‖λ ≤ cλ,μNλ−μ‖u‖μ, λ ≤ μ (Sobolev norms of periodic functions) in O(N log N) arithmetical operations.

We present an existence result for a partial differential inclusion with linear parabolic principal part and relaxed one-sided Lipschitz multivalued nonlinearity in the framework of Gelfand triples. Our study uses discretizations of the differential inclusion by a Galerkin scheme, which is compatible with a conforming finite element method, and we analyze convergence properties of the discrete ...

The problem of interpolating between discrete fields arises frequently in computational physics. The obvious approach, consistent interpolation, has several drawbacks such as suboptimality, non-conservation, and unsuitability for use with discontinuous discretisations. An alternative, Galerkin projection, remedies these deficiencies; however, its implementation has proven very challenging. This...

We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretiz...

We analyze a fully discrete spectral method for the numerical solution of the initial-and periodic boundary-value problem for two nonlinear, non-local, dispersive wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The equations are discretized in space by the standard Fourier-Galerkin spectral method and in time by the explicit leapfrog scheme. For the resulting fully di...