An efficient Haar wavelet method for the coupled non-linear transient PDE-ODEs system with discontinuous coefficients

TL;DR

This paper introduces a Haar wavelet method tailored for efficiently solving coupled non-linear transient PDE-ODE systems with discontinuous coefficients, demonstrating high accuracy and computational efficiency across multiple dimensions.

Contribution

The paper presents a novel Haar wavelet approach capable of handling jump discontinuities in coupled PDE-ODE systems, with proven convergence and improved solver acceleration techniques.

Findings

Method handles multiple discontinuities effectively.

Convergence is exponential order.

ILU-GMRES accelerates convergence more than other solvers.

Abstract

In this work, the Haar wavelet method for the coupled non-linear transient PDE-ODEs system with the Neumann boundary condition has been proposed. The capability of the method in handling multiple jump discontinuities in the coefficients and parameters of the transient and coupled PDE-ODEs system is brought out through a series of 1D/2D/3D test problems. The method is easy to implement, computationally efficient, and compares well with conventional methods like the finite element method. Convergence analysis of the method has been carried out and apriori error is found to be exponential order i.e. ||u-u_H||_X<= o(2^j ). ILU-GMRES is found to better accelerate the numerical solution convergence than other Krylov solvers for the class of transient non-linear PDE-ODE system.