A high-precision method for general nonlinear initial-boundary value problems

TL;DR

This paper introduces a high-precision, space-time decoupled wavelet numerical method for nonlinear initial-boundary value problems, achieving high order accuracy and stability, with demonstrated superior efficiency over existing methods.

Contribution

A novel wavelet-based numerical scheme using Coiflet functions for high-order, fully decoupled space-time solution of nonlinear boundary value problems, including stability analysis.

Findings

Achieves Nth-order accuracy with Coiflet wavelets.

Demonstrates superior accuracy and efficiency in numerical tests.

Provides a stable and explicit solution framework for nonlinear problems.

Abstract

A high precision, and space time fully decoupled, wavelet formulation numerical method is developed for a class of nonlinear initial boundary value problems. This method is established based on a proposed Coiflet based approximation scheme with an adjustable high order for a square integrable function over a bounded interval, which allows expansion coefficients to be explicitly expressed by function values at a series of single points. In applying the solution method, the nonlinear initial boundary value problems are first spatially discretized into a nonlinear initial value problem by combining the proposed wavelet approximation scheme and the conventional Galerkin method. A novel high order step by step time integrating approach is then developed for the resulting nonlinear initial value problem using the same function approximation scheme based on wavelet theory. The solution method is shown to have Nth-order accuracy, as long as the Coiflet with [0, 3N-1] compact support is adopted, where N can be any positive even number. In addition, the stability property of the method is analyzed, and the stable domain is determined. Numerical examples are considered to justify both the accuracy and efficiency of the method. Results show that the proposed solution method has better accuracy and efficiency than most other methods.