Computational algorithms for modeling systems with piecewise constant parameters

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.


Introduction
Numerical modelling of systems in which the parameters characterizing the physical properties of the medium are piecewise constant and, therefore, abruptly change when crossing the interface between the parts of the system, is undoubtedly an urgent problem and occurs in a variety of applied areas. For example, when simulating the propagation of elastic or electromagnetic disturbances in layered media or media consisting of parts with different physical properties, the behaviour of elastically supported bearing structures consisting of parts with different properties of the materials used, heat propagation in composite materials, and many others. Also, it is a very common technique when, in mathematical modelling, the system is divided into subdomains, within each such subdomain, the system parameters are assumed to be constant and abruptly change when crossing the boundary separating the parts. We would especially like to highlight the case when the system is under the action of external forces, which are transmitted through a small in comparison with the size of the system "contact patch", such as in the case of a bearing structure, supported on elastic supports. This system also belongs to the considered ones  the elastic properties of the supports on which the structure is located change abruptly.
At the boundaries separating the parts of the system, at which the parameters of the system change abruptly, the smoothness conditions for the variables of the mathematical model may be violated. These jumps of variables or their derivatives cannot be arbitrary but must obey conditions reflecting the physical conditions at the interface between the media. For example, there must be a continuous flow of heat through the boundary of the media, or Newton's third law must be fulfilled the force with which one of the contacting media acts on the other is equal and oppositely directed to the force with which the second acts on the first. These physical conditions must also be met for the model variables. Otherwise, the conservatism of the model will be violated, and the numerical model will not be adequate to the investigated physical situation.
One of the most frequently used algorithms in the numerical simulation of systems with piecewise constant parameters is the semi-discrete Galerkin method and its various modifications, in which the physical conditions at the interface between media are included in the model equations in a "weak" form [1]- [7]. This approach makes it possible to approximately satisfy the conditions at the interface between the media with an accuracy corresponding to the accuracy of the numerical model.
A feature of the Galerkin method is that it leads to implicit computational algorithms at each time step, it is necessary to solve a system of algebraic equations, which significantly reduces the speed.
In this work, on the example of the problem of vibrations of an elastic string supported on intermediate supports, we propose a unified approach to the numerical modelling of systems with piecewise constant parameters, which allows one to construct explicit numerical algorithms. This approach is based on the formulation of the initial-boundary value problem in the form of a system of partial differential equations for generalized functions, in the construction of the fundamental solution of the operator of the problem and the representation of the solution of the problem as a compression of the fundamental solution with the right-hand side of the system of equations for generalized functions.

Problem statement
The methods for constructing numerical models outlined below will be demonstrated by the example of a model of vibration of an elastic string with free ends, supported on several intermediate elastic supports, each of which is in contact with the string through a small "contact patch".
Let us denote   reflecting the condition of string continuity, and the condition reflecting the fulfillment of Newton's third law (the forces acting on the boundary from both sides are equal in magnitude and oppositely directed).

Generalized Riemann problem with additional conditions on the boundary
In the further presentation, we will use the concepts and statements of the theory of generalized functions of slow growth, a presentation of which can be found, for example, in [5] , and the functional f acts on the base vector-function The algorithms proposed in this work are based on the solution of the generalized Riemann problem with additional conditions on the boundary. In papers [4], [5], [8], the statement and solution of the generalized Riemann problem with additional conditions on the boundary are given as applied to systems of partial differential equations of the first order. Below is a solution to the generalized Riemann problem for equations with second-order derivatives.
In the application to the problem under consideration, the generalized Riemann problem with additional conditions on the boundary is formulated as follows. To find a solution to the Cauchy problem for the equation

Equation coefficients are constant in the left and right half-planes
are everywhere twice continuously differentiable, except for the point 0 x  . A solution of the Cauchy problem at every moment of time 0 t  at the point 0 x   must satisfy conjugation conditions (2) and (3).
Let the function   , u t x is a solution to the Cauchy problem (4). Let us plot the functions   at 0, 0 xt  and are equal to zero otherwise. Then let us construct functions and equal to zero otherwise. We also denote by    the value of the solution and its derivative, respectively, on the right and left sides of the boundary.
The function   , u t x  considered as a generalized function from  satisfies the equation Let us introduce the notation Then the solution to equation (5) can be written as a compression of the fundamental solution with the right-hand side Considering that  Passing in equalities (11) to the limit at 0 x  from the left, we obtain Passing in equalities (13) to the limit at 0 x  from the right, we obtain Let us multiply the first equation in (12) Let us multiply the second equation in (12) by   and add to it the second equation from (14) multiplied by   . Let us take into account the conjugation conditions (2) Relations (17) and (18) will be used in what follows when constructing computational algorithms. Also, when constructing a computational algorithm, the solution of a somewhat different problem will be used, which we will call the generalized Riemann problem with conditions on the outer boundary, and which is formulated as follows. Find a solution in the left-hand side-plane 0 x  of the initial-boundary value problem for a partial differential equation with constant coefficients that satisfies the free boundary conditions We obtain a solution to this problem similarly to the above solution to the generalized Riemann problem with conditions on the internal boundaries.
Let the function   , u t x be a solution to problem (19). Let us construct functions , at 0, 0 xt  and are equal to zero otherwise. We also denote by Passing in equality (21) Relations (24), along with relations (17) and (18), will be used in the further presentation when constructing computational algorithms.    (17) and (18), as a solution to the generalized Riemann problem with conditions on the inner boundary. And the same values for 1, iI  are determined by expressions (24), as a solution to the generalized Riemann problem with conditions on the outer boundary.

Formulation of the problem in generalized functions
These equations, using matrix notation, can be rewritten At all   , tx   φS the equality is true Taking into account the initial conditions (1), we get