Modified Difference Scheme Research Articles

Numerical approaches for solution of hyperbolic difference equations on circle

Abstract The present paper considers nonlocal boundary value problems for hyperbolic equations on the circle T 1 \mathbb{T}^{1} . The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided.

An approximation of problems of optimal control on the coefficients of elliptic convection-diffusion equations with an imperfect contact matching condition

We consider nonlinear optimization problems for processes described by non-self-adjoint elliptic equations of convection-diffusion problems with an imperfect contact matching conditions. These are the problems with a jump of the coefficients and of the solution on the interface; the jump of the solution is proportional to the normal component of the flux. Variable coefficients multiplying the highest and the lowest derivatives in the equation and the coefficients by nonlinear terms in the equations of state are used as controls. Finite difference approximations of optimization problems are constructed and investigated. For the approximation of state equations we propose a new ``modified difference scheme'' in which the variable grid coefficients in the principal part of the difference operator are computed using method other than traditionally applied in the theory of difference schemes. The problem's correctness is investigated. The accuracy estimation of difference approximations with respect to the state are obtained. Convergence rate of approximations with respect to cost functional is estimated, too. Weak convergence with respect to control is proved. The presence of a non-self-adjoint operator causes certain difficulties in constructing and studying approximations of differential equations describing discontinuous states of controlled processes, in particular, in proving the difference approximations well-posedness, and in studying the relationship between the original optimal control problem and the approximate mesh problem. The approximations are regularized. The obtained results will be heavily used later in solving problems associated with the development of effective methods for the numerical solution to the constructed finite-dimensional mesh optimal control problems and their computer implementation.

Existence and stability of solutions to the compressible Euler equations with an outer force

We study the compressible Euler equation with an outer force. The global existence theorem has been proved in many papers, provided that the outer force is bounded. However, the stability of their solutions has not yet been obtained until now. Our goal in this paper is to prove the existence of a global solution without such an assumption as boundedness. Moreover, we deduce a uniformly bounded estimate with respect to the time. This yields the stability of the solution.When we prove the global existence, the most difficult point is to obtain the bounded estimate for approximate solutions. To overcome this, we employ an invariant region, which depends on both space and time variables. To use the invariant region, we introduce a modified difference scheme. To prove their convergence, we apply the compensated compactness framework.

An Operator‐Difference Method for Telegraph Equations Arising in Transmission Lines

A second‐order linear hyperbolic equation with time‐derivative term subject to appropriate initial and Dirichlet boundary conditions is considered. Second‐order unconditionally absolutely stable difference scheme in (Ashyralyev et al. 2011) generated by integer powers of space operator is modified for the equation. This difference scheme is unconditionally absolutely stable. Stability estimates for the solution of the difference scheme are presented. Various numerical examples are tested for showing the usefulness of the difference scheme. Numerical solutions of the examples are provided using modified unconditionally absolutely stable second‐order operator‐difference scheme. Finally, the obtained results are discussed by comparing with other existing numerical solutions. The modified difference scheme is applied to analyze a real engineering problem related with a lossy power transmission line.

Modified technique for solving advance-delay differential systems

In this paper, the numerical treatment for two nonlinear advance delay differential system is discussed. This system arises during the study of conduction of impulses down myelinated axons, incorporating electrical properties of the myelin. This system consists of two coupled first-order equations, one having a delay term and the other having an advance one. The modified difference scheme is used and then applied to some test problems.

Analysis of thin layers and discontinuities

A modification of the finite difference scheme for thin dielectric layers is presented and implemented into the method of lines. The thin layers may be placed between two consecutive discretisation lines, i.e. no line is inside the layer. The analysis is based on the treatment of dielectric boundaries. So, formulas for the finite difference scheme at interfaces are given, too. The expressions are tested by determining the effective index of a slab waveguide with a very thin film-layer. A comparison with the analytical solution proves the accuracy of the derived formulas. Another test was the analysis of a Bragg-grating, here only the treatment of interfaces was checked. The modified difference scheme works very well in the TE-case whereas there remain problems for the TM-polarisation.

Numerical approximations in pile‐driving analysis

AbstractA finite‐difference scheme widely used in analysis of pile driving is examined, the scheme is shown to yield peculiar results in simple symmetric impact problems and to inherently posses an imprecise definition of energy balance. A modified difference scheme is suggested and comparisons of the two methods are shown.