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Exponential Euler and backward Euler methods for nonlinear heat conduction problems

M. A. Botchev, V. T. Zhukov

TL;DR

The paper addresses nonlinear heat conduction problems arising from spatial discretization, formulated as $y'(t) = -A(y(t))\,y(t) + g(t)$. It compares two temporal discretizations: a backward Euler method with nonlinear inner iterations and a nonlinear exponential Euler scheme, proving monotonicity and boundedness for both and establishing convergence conditions for the nonlinear solves. The exponential Euler approach uses a Krylov-based evaluation of $\varphi(-\Delta t A_m)$ to advance the solution, rendering the method essentially explicit. Numerical tests in 1D and 2D demonstrate similar stability and accuracy for both schemes, with the exponential Euler typically requiring fewer matrix-vector products than the LI-M (local-iteration modified) approach, highlighting its computational efficiency.

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).

Exponential Euler and backward Euler methods for nonlinear heat conduction problems

TL;DR

The paper addresses nonlinear heat conduction problems arising from spatial discretization, formulated as . It compares two temporal discretizations: a backward Euler method with nonlinear inner iterations and a nonlinear exponential Euler scheme, proving monotonicity and boundedness for both and establishing convergence conditions for the nonlinear solves. The exponential Euler approach uses a Krylov-based evaluation of to advance the solution, rendering the method essentially explicit. Numerical tests in 1D and 2D demonstrate similar stability and accuracy for both schemes, with the exponential Euler typically requiring fewer matrix-vector products than the LI-M (local-iteration modified) approach, highlighting its computational efficiency.

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).
Paper Structure (8 sections, 5 theorems, 41 equations, 2 figures, 2 tables)

This paper contains 8 sections, 5 theorems, 41 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and problem setting
  2. Solution methods
  3. The backward Euler method
  4. Nonlinear exponential Euler scheme
  5. Numerical experiments
  6. 1D heat equation
  7. 2D heat equation
  8. Conclusions and an outlook to further research

Key Result

Corollary 1

Solution $y(t)$ of the semidiscrete IVP ivp is entrywise nonnegative, i.e., provided that conditions aij and g>=0 hold.

Figures (2)

  • Figure 1: Numerical Green functions of the backward Euler (left) and exponential Euler (right) methods at time $T=0.1$ computed with time steps $\Delta t=T$ (top) and $\Delta t=T/1000$ (bottom) on a uniform grid with $N=128$ nodes.
  • Figure 2: Solution (left) and error (right) of the exponential Euler scheme on the grid $N=128$ at $T=0.5$, computed with the time step size $\Delta t=10^{-3}$ ($\Delta t \max_{t\in[0,T]}\|A(y(t))\|_1\approx 65$).

Theorems & Definitions (12)

  • Corollary 1
  • proof
  • Remark 1
  • Corollary 2
  • proof
  • Lemma 1
  • proof
  • Corollary 3
  • proof
  • Corollary 4
  • ...and 2 more