Characteristic directions approach to solving scalar one-dimensional nonlinear advection equation with non-convex flow function

Abstract

A concept of the characteristic technique used to obtain a generalized solution of the scalar one-dimensional nonlinear advection equation with the non-convex flow function is presented. Two grids: characteristic and Eulerian are used to obtain numerical solution. A characteristic grid is adaptive both to the properties of the initial distribution function and to the properties of the boundary condition function. This allows: development of the algorithm for obtaining a numerical solution on characteristic grid using the properties of the solution of nonlinear advection equation in smooth region; to reproduce spatial location and solution value at the discontinuity points and extreme points at the accuracy determined by interpolation and approximation of initial values and boundary condition functions. For the non-convex flow function, algorithms are proposed for the definition of the sequence of Riemann problems (strong discontinuity) and for their solving. Refined expressions are derived for the velocity of a strong non-stationary discontinuity. Construction of the solution with satisfying of integral preservation law for the non-convex flow function is presented. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)