Class Of Difference Schemes Research Articles

Numerical benchmark for the charge cycle in a combustion engine

For the modeling and numerical solution of the compressible flow in a benchmark engine a network formulation is used. The presented engine consists of a single cylinder with inlet and outlet valve, respectively one suction and exhaust pipe and two connections with the atmosphere. The time history of the cylinder is described by an ODE system which introduces heat losses and combustion energy by means of measurement. Considering the gas flow in pipes leads to the unsteady, one-dimensional Euler equations which are numerically treated by a TVD scheme. In contrast to classical difference schemes spurious numerical oscillations are prevented. A DAE system in time is achieved by a semi-empirical approach for the cylinder and its connections, and it is solved by a predictor-corrector method. As numerical experiments show, the presented approach results in a fast solver for the benchmark engine and is of interest in industrial research too.

Thermodynamically matched difference schemes

A class of difference schemes which approximate the system of one-dimensional non-stationary equations of gas dynamics in Lagrange mass variables is considered. The homogeneous difference schemes satisfy the scheme matching principle and the equation of state of an ideal gas. The matching is based on supplementary laws of conservation which are a consequence of the equations of gas dynamics and the equations of state. The efficiency of the difference schemes is confirmed by the results of numerical computations of a model problem.

The uniformly convergent difference schemes for a singular perturbation problem of a self-adjoint ordinary differential equation

In this paper, we construct a class of difference schemes with fitted factors for a singular perturbation problem of a self-adjoint ordinary differential equation. Using a different method from [1], by analyzing the truncation errors of schemes, we give the sufficient conditions under which the solution of the difference scheme converges uniformly to the solution of the differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence, and applying them to example, obtain the numerical results.

Uniform convergence for the difference scheme in the conservation form of ordinary differential equation with a small parameter

In this paper, we consider a singular perturbation boundary problem for a self-adjoint ordinary differential equaiton. We construct a class of difference schemes with fitted factors, and give the sufficient conditions under which the solution of difference scheme converges uniformly to the solution of differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence.

$L_p $-Estimates of Difference Schemes for Strictly Hyperbolic Systems with Nonsmooth Data

Lower bounds for the growth in $L_p $ of powers of difference schemes for strictly hyperbolic systems are established. In particular, it is shown that schemes with order of accuracy $\mu $ and of order dissipativity s, with $\mu + 1 < s$, in a sense are more unstable than the solution of the corresponding continuous problem indicates. We also provide bounds on the $l_{p'} $-norm over the mesh of the discrete (approximate) solution in terms of $l_p $ -norms of the data, with $1 \leqq p \leqq 2$, ${1 / p} + {1 / {p'}} = 1$. Besides showing that such estimates considerably improve on the corresponding $l_{p'} - l_{p'} $-estimates for classes of data likely to appear in applications (cf. [2]), we also exhibit a class of difference schemes which take maximal advantage of this property.

Difference schemes with best possible truncation error

In this paper we present a variety of schemes for solving the initial value problem for a class of hyperbolic systems of partial differential equations. These schemes arise as solutions of constrained minimization problems for a quadratic form. The form is an expression for the local truncation error for a certain class of difference schemes.