Adaptive Multiple Knot B-spline Wavelets for Solving Saint-Venant equations

Solving the Saint-Venant equations by numerical methods like finite element and finite difference methods yields an unstable solution for a fairly large open channel. Multiple Knot B-Spline Wavelets (MKBSW) are in the class of semi-orthogonal wavelets that have compact support. Hence, these basis functions are suitable for solving the Saint-Venant equations. However, solving the Saint-Venant equations by MKBSW method requires a long CPU time. In this paper, we present an adaptive wavelet method to solve the Saint-Venant equation in a fairly short time. In fact, we first solve the problem in a few first moments and then by statistical methods of time series and regression, where the active wavelets are predicted in the next moments. Moreover, by this adaptive method, the cumulative errors (that are produced by solving the discretized system, numerically) are decreased for large open channels. Two numerical examples are given to support our results.