The Maximum Principle for a Hyperbolic Grid System and Its Applications

Abstract

The extremum principle for one hyperbolic equation [namely, system (1) with n = 1] under some constraints on the coefficients and the solution class [1] was proved for the first time in [1]. The maximum principle for the absolute value |U(ξ, η)| = ( ∑n i=1 ui(ξ, η) 2) was studied in [2–4] for system (1) in some special cases. Important new extremal properties of solutions of system (1) were obtained in [5, 6]. In the present paper, we prove the maximum principle for the grid analog of system (1) and use it to prove the existence and uniqueness of the solution of the finite-difference counterpart of the Darboux problem. The finite-difference method was earlier used for the approximate solution of boundary value problems in [7–10]. To construct the grid domain, we split the interval AB into N = 2N1 equal parts and draw characteristics of system (1) through the partition points. We obtain a grid of square cells with mesh width h. The set of grid points belonging to ∆ (respectively, ∆) will be denoted by ∆h (respectively, ∆h). We number the points of ∆h as follows: first, we number the points lying on AC in ascending order of η : (0, 0), (0, 1), (0, 2), . . . , (0, N); then we number the points lying on the line ξ = h in the same order: (1, 1), (1, 2), . . . , (1, N), and so on. Throughout the following, we assume that the coefficients of system (1) and the functions fi are continuous in ∆. The value of a function at a grid point (k,m) is understood as the value at the point with coordinates (kh,mh). System (1) can be approximated at a grid point (k,m) ∈ ∆h by the system of finite-difference equations RiU(k,m) ≡ [(1 + aih+ bih) ui(k,m)− (1 + aih) ui(k − 1,m) − (1 + bih) ui(k,m− 1) + ui(k − 1,m− 1)]/h