Search
Results
-
An Accuracy Barrier for Stable Three-Time-Level DifferenceSchemes for Hyperbolic Equations
(1997)SAM Research ReportWe consider three-time-level difference schemes for the linear constant coefficient advection equation $u_t$ = $cu_x$. In 1985 it was conjectured that the varrier to the local order $p$ of schemes which are stable is giben by $p \le $2 min {$R$,$S$}. Here $R$ and $S$ denote the number of downwind and upwind points, respectively, in the difference stencil with respect to the characteristic of the differential equation through the update ...Report -
Accurancy barriers of three time level difference schemes for hyperbolic equations
(1991)SAM Research ReportA basic assumption for the interior scheme when solving hyperbolic mixed initial boundary value problems is that it satisfies the von Neumann stability condition. Here we show that this condition limits the order of accuracy a scheme with a given difference stencil can have. The proofs use order stars.Report -
Special aspects of reacting inviscid blunt body flow
(1992)SAM Research ReportThe problem of a hypersonic blunt body flow in two space dimensions is considered. The governing inhomogeneous Euler equations are given and the special treatment of calorically non-perfect gas in chemical non-equilibrium is described. The chemical model is given. The problem of the arising chemical boundary layer is discussed. Analytical and numerical investigations are used to analyse this boundary layer and to get error estimates for ...Report -
A new multidimensional Euler scheme
(1992)SAM Research ReportA new idea is presented to solve the multidimensional Euler equations numerically. The aim of this idea is to obtain a robust shock capturing method without the use of dimensional splitting. The starting point is the idea of the one-dimensional flux vector splitting and the homogeneity of the Euler equations. Using this concept it is shown that a different interpretation of the one-dimensional waves and the use of some physical properties ...Report -
The method of transport for nonlinear systems of hyperbolic conservation laws in several space dimensions
(1997)SAM Research ReportThe idea of the method of transport is introduced by means of non-linear conservation laws. The systems are rewritten in an advection form accounting for the characteristic propagation directions. A straightforward linearization of this advection form leads to a genuinely multi-dimensional method. Approximations using infinitely many or a finite number of propagation directions are shown.Report -
An Accuracy Barrier for Stable Three-Time-Level Difference Schemes for Hyperbolic Equations
(1995)SAM Research ReportWe consider three-time-level difference schemes for the linear constant coefficient advection equation $u_t = cu_x$. In 1985 it was conjectured that the varrier to the local order $p$ of schemes which are stable is giben by $p \le 2$ min {$R$,$S$}. Here $R$ and $S$ denote the number of downwind and upwind points, respectively, in the difference stencil with respect to the characteristic of the differential equation through the update ...Report -
Stability of time discretization, Hurwitz determinants and order stars
(1995)SAM Research ReportWe shall review stability requirements for time discretizations of ordinary and partial differential equations. If a constant time step is used and the method involves more than two time levels stability is always related to the location of roots of a polynomial in circular or half plane regions. In several cases the coefficients of the polynomial depend on a real or complex parameter. Hurwitz determinants allow to create a fraction free ...Report -
The influence of a source term, an example: chemically reacting hypersonic flow
(1992)SAM Research ReportWe show, by way of an example, that the solution of a system of hyperbolic conservation laws exhibits an unexpected behavior if a source term is present. The example is the system of Euler equations for N species in two space dimensions. If the source term is not present and in the initial and inflow conditions a fixed mixture of species is prescribed then the solution basically behaves like the flow of an ideal gas, except that there are ...Report -
Error Estimators for the Position of Discontinuities in Hyperbolic Conservation Laws with Source Terms which are solved using Operator Splitting
(1997)SAM Research ReportWhen computing numerical solutions of hyperbolic conservation laws with source terms, one may obtain spurious solutions --- these are unphysical solutions that only occur in numerics such as shock waves moving with wrong speeds, cf. [7], [2], [1], [10], [3]. Therefore it is important to know how errors of the location of a discontinuity can be controlled. To derive appropriate error-estimates and to use them to control such errors, is the ...Report -
A numerical method for unsteady flows
(1994)SAM Research ReportA high resolution finite volume method for the computation of unsteady solutions of the Euler equations in two space dimensions is presented and validated. The scheme is of Godunov-type. The first order part of the flux function uses the approximate Riemann problem solver of Pandolfi and here a new derivation of this solver is presented. This construction paves the way to understand the conditions under which the scheme satisfies an entropy ...Report