Computational algorithms for modeling systems with piecewise constant parameters

Abstract

Modeling systems with piecewise constant parameters imposes special requirements on the computational algorithms used. In places where the parameters are discontinuous, the model variables may experience discontinuities or smoothness disturbances. In this case, the conditions for matching the values of the variables on both sides of the discontinuity boundary, reflecting the physical conditions at this boundary, must be satisfied. The paper proposes computational algorithms for modeling these systems, based on the formulation of the initial initial-boundary value problem in the form of a system of partial differential equations for generalized functions. In this case, the initial conditions, conditions on the external and internal boundaries are included in these equations in a weak form. The latter circumstance makes it possible to avoid the need to subordinate the basis functions, according to which the desired solution is expanded, to the conditions on the outer and inner boundaries, which is an important circumstance, especially for problems with many spatial variables. The paper presents a solution to the generalized Riemann problem with conditions on the outer or inner boundary for a differential equation with second-order derivatives with respect to spatial variables. The solution to the Riemann problem is based on the formulation of the problem in the form of a partial differential equation for generalized functions and the construction of a fundamental solution to the problem operator. When constructing a computational algorithm, the solution of the generalized Riemann problem is used on the inner and outer boundaries. The proposed technique allows, by using high-degree polynomials as basis functions, to build computational algorithms of a high order of approximation.