Exponential Euler and backward Euler methods for nonlinear heat conduction problems

TL;DR

This paper introduces a nonlinear exponential Euler scheme for heat conduction problems, compares it with backward Euler, and demonstrates its efficiency and convergence properties through numerical tests.

Contribution

It proposes a novel exponential Euler method for nonlinear heat problems and compares its performance with backward Euler, highlighting advantages in efficiency and implementation.

Findings

The exponential Euler scheme is essentially explicit and efficient.

Both methods show monotonicity and boundedness under certain conditions.

Numerical tests confirm the effectiveness of the exponential Euler scheme.

Abstract

In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved. We compare this method to the backward Euler method combined with nonlinear iterations. For both methods we show monotonicity and boundedness of the solutions and give sufficient conditions for convergence of the nonlinear iterations. Numerical tests are presented to examine performance of the two schemes. The presented exponential Euler scheme is implemented based on restarted Krylov subspace methods and, hence, is essentially explicit (involves only matrix-vector products).