The Galerkin-Implicit Method for Solving Nonlinear Variable Coefficients Hyperbolic Boundary Value Problem

Jamil A Ali Al-Hawasy,Nuha Farhan Mansour

doi:10.24996/ijs.2021.62.11.20

Abstract

This paper has the interest of finding the approximate solution (APPS) of a nonlinear variable coefficients hyperbolic boundary value problem (NOLVCHBVP). The given boundary value problem is written in its discrete weak form (WEFM) and proved have a unique solution, which is obtained via the mixed Galerkin finite element with implicit method that reduces the problem to solve the Galerkin nonlinear algebraic system (GNAS). In this part, the predictor and the corrector techniques (PT and CT, respectively) are proved at first convergence and then are used to transform the obtained GNAS to a linear GLAS . Then the GLAS is solved using the Cholesky method (ChMe). The stability and the convergence of the method are studied. Some illustrative examples are used, where the results are given by figures that show the efficiency and accuracy for the method.

Highlights

Hyperbolic partial differential equations arise in many physical problems, such as vibrating strings, and in many other fields such as fluid dynamics, optics, and others
The given boundary value problem is written in its weak form (WEFM), and it is discretized using the mixed Galerkin finite element method (GFEME) for the space variable with the implicit method (IM) for the time variable (MGFEIM)
By setting in the WEFM of (5-7), it is discretized by using the GFEME as follows: let the domain is divided into sub regions

Summary

1.Introduction

Hyperbolic partial differential equations arise in many physical problems, such as vibrating strings, and in many other fields such as fluid dynamics, optics, and others. The finite element method has been studied by many researchers who are interested in this field to solve LHBVP. In. 2018, Wick studied in his book the GFEME for solving LHBVP and NOLHBVP with constant coefficients [7]. The given boundary value problem is written in its WEFM, and it is discretized using the mixed Galerkin finite element method (GFEME) for the space variable with the implicit method (IM) for the time variable (MGFEIM). 3. Assumptions (i) Let κ1 and κ2 be two positive constants such that the following are satisfied: a)⎹. By setting in the WEFM of (5-7), it is discretized by using the GFEME as follows: let the domain is divided into sub regions.

Stability

Cholesky Factorization