Approximate Solutions and Error Bounds for Quasi-Linear Hyperbolic Initial Boundary Value Problems

Abstract

For a given quasilinear hyperbolic initial boundary value problem whose nonlinear part is convex, constrained minimization problems (CMP) are formulated, making use of a minimum principle of the hyperbolic equation. Approximate solutions and two error bounds of the given initial boundary value problem can be obtained by solving these CMP’s, using linear programming methods and numerical approximation techniques. The CMP’s are iterative processes. Each iterative cycle consists of two steps. Starting with some guessed approximation, Step 1 improves the solution by solving a linear program whose constraints are quasi-linearizations of the given problem and whose objective function is a linear function of the defects of the constraints. Step 2 is also a linear program, whose constraints are an auxiliary system of the given problem. Error bounds are formed from the solutions of Step 1 and Step 2. The approximate solutions are obtained by minimizing the error bounds in a certain sense. Numerical results are included.