Julia Novo

Linearly implicit Runge—Kutta methods for advection—reaction—diffusion equations

Abstract We construct two variable-step linearly implicit Runge–Kutta methods of orders 3 and 4 for the numerical integration of the semidiscrete equations arising after the spatial discretization of advection–reaction–diffusion equations. We study the stability properties of these methods giving the appropriate extension of the concept of L -stability. Numerical results are reported when the methods presented are combined with spectral discretizations. Our experiments show that the methods, being easily implementable, can be competitive with standard stiffly accurate time integrators.

Postprocessing the Galerkin Method: a Novel Approach to Approximate Inertial Manifolds

A postprocess of the standard Galerkin method for the discretization of dissipative equations is presented. The postprocessed Galerkin method uses the same approximate inertial manifold

An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equation

\Phi_{app}

Error Analysis of the SUPG Finite Element Discretization of Evolutionary Convection-Diffusion-Reaction Equations

to approximate the high wave number modes of the solution as in the nonlinear Galerkin (NLG) method. However, in this postprocessed Galerkin method the value of

On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations

\Phi_{app}

The Postprocessed Mixed Finite-Element Method for the Navier--Stokes Equations

is calculated only once, after the time integration of the standard Galerkin method is completed, contrary to the NLG in which

A Spectral Element Method for the Navier--Stokes Equations with Improved Accuracy

\Phi_{app}

A postprocessed Galerkin method with Chebyshev or Legendre polynomials

evolves with time and affects the time evolution of the lower wave number modes. The postprocessed Galerkin method, which is much cheaper to implement computationally than the NLG, is shown, in the case of Fourier modes, to possess the same rate of convergence (accuracy) as the simplest version of the NLG, which is based on either the Foias--Manley--Temam approximate inertial manifold or the Euler--Galerkin approximate inertial manifold. This is proved for some problems in one and two spatial dimensions, including the Navier--Stokes equations under periodic boundary conditions. The advantages of postprocessing that we present here apply not only to the standard Galerkin method, but also to the computationally more efficient pseudospectral method.

Efficient methods using high accuracy approximate inertial manifolds

In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to standard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary conditions. Specifically, we study in detail the two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations.

A postprocess based improvement of the spectral element method

Conditions on the stabilization parameters are explored for different approaches in deriving error estimates for the streamline-upwind Petrov-Galerkin (SUPG) finite element stabilization of time-dependent convection-diffusion-reaction equations. Exemplarily, it is shown for the SUPG method combined with the backward Euler scheme that standard energy arguments lead to estimates for stabilization parameters that depend on the length of the time step. The stabilization vanishes in the time-continuous limit. However, based on numerical experience, this seems not to be the correct behavior. For this reason, the main focus of the paper consists in deriving estimates in which the stabilization parameters do not depend on the length of the time step. It is shown that such estimates can be obtained in the case of time-independent convection and reaction. An error estimate for the time-continuous case with the standard order of convergence is derived for stabilization parameters of the same form as they are optimal for the steady-state problem. Analogous estimates are obtained for the fully discrete case using the backward Euler method and the Crank-Nicolson scheme. Numerical studies support the analytical results.