Miloslav Vlasák

Time discretizations for evolution problems

The aim of this work is to give an introductory survey on time discretizations for liner parabolic problems. The theory of stability for stiff ordinary differential equations is explained on this problem and applied to Runge-Kutta and multi-step discretizations. Moreover, a natural connection between Galerkin time discretizations and Runge-Kutta methods together with order reduction phenomenon is discussed.

A Posteriori Error Estimates for Nonstationary Problems

We apply continuous and discontinuous Galerkin time discretization together with standard finite element method for space discretization to the heat equation. For the numerical solution arising from these discretizations we present a guaranteed and fully computable a posteriori error upper bound. Moreover, we present local asymptotic efficiency estimate of this bound.

Implicit–Explicit Backward Difference Formulae Discontinuous Galerkin Finite Element Methods for Convection–Diffusion Problems

We deal with a numerical solution of a scalar nonstationary convection–diffusion equation with a nonlinear convection and a linear diffusion. We carry out the space semi-discretization with the aid of the symmetric interior penalty Galerkin (SIPG) method and the time discretization by backward difference formulae (BDF) and suitable linearization of nonlinear convective term. The resulting scheme is unconditionally stable, has a high order of accuracy with respect to space and time coordinates and requires solutions of linear algebraic problems at each time step. We derive a priori error estimates in the L ∞ (L 2)-norm up to the order 6 in time.

Analysis of Semi–Implicit Runge‐Kutta‐DGFEM for a Semilinear Convection‐Diffusion Equation

We deal with the numerical solution of a scalar semilinear convection‐diffusion equation, which represents a simplified model to the compressible Navier‐Stokes equations. For discretization in space we use the discontinuous Galerkin finite element method (DGFEM) which is a promising scheme for solving problems with shocks and boundary layers. For the time discretization we use implicit Runge–Kutta methods to obtain the scheme which should be able to solve stiff problems in combination with some suitable explicit linearization for treatment of nonlinear terms. The resulting scheme is sufficiently robust and needs only solution of a linear problem at each time step. We present a priori error estimate.

Semi‐Implicit Solution of the Convection‐Diffusion Problems with Applications in Fluid Dynamics

We deal with the time discretization of the system of ordinary differential equations arising from the discontinuous Galerkin discretization of a scalar convection‐diffusion equation. We discuss several semi‐implicit techniques, which lead to a sufficiently stable scheme and to a necessity to solve only linear problem at each time level. An extension to the system of the Navier‐Stokes equations is presented and demonstrated by a numerical example.