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Uniform estimates for a fully discrete scheme integrating the linear heat equation on a bounded interval with pure Neumann boundary conditions

Guillaume Dujardin, Pauline Lafitte

TL;DR

The paper addresses uniform-in-time error estimation for a fully discrete scheme solving the linear heat equation with Neumann boundary conditions on a bounded interval. It combines symmetric finite-difference spatial discretization with explicit Euler time stepping under a CFL condition and leverages a spectral analysis of the discrete Laplacian, together with a careful decomposition of boundary-induced consistency defects, to prove a uniform convergence rate of order $O( ext{δx})$ (and $O( ext{δx}+ ext{δt})$ in a Corollary) for the homogeneous problem, extended to the nonhomogeneous case and to steady-state computation. A detailed error analysis partitions the error into propagation and consistency components, analyzes boundary terms via a spectral decomposition, and establishes uniform-in-time bounds through discrete Gronwall-type arguments and exponential decay properties. The work also shows how to compute steady states by time-stepping a nonhomogeneous heat equation and proves corresponding uniform-in-time error bounds, with extensions to higher dimensions demonstrated numerically. Overall, the results provide a robust framework for long-time integration of parabolic problems with Neumann boundary conditions and offer practical tools for steady-state computations in both 1D and 2D settings.

Abstract

This manuscript deals with the analysis of numerical methods for the full discretization (in time and space) of the linear heat equation with Neumann boundary conditions, and it provides the reader with error estimates that are uniform in time. First, we consider the homogeneous equation with homogeneous Neumann boundary conditions over a finite interval. Using finite differences in space and the Euler method in time, we prove that our method is of order 1 in space, uniformly in time, under a classical CFL condition, and despite its lack of consistency at the boundaries. Second, we consider the nonhomogeneous equation with nonhomogeneous Neumann boundary conditions over a finite interval. Using a tailored similar scheme, we prove that our method is also of order 1 in space, uniformly in time, under a classical CFL condition. We indicate how this numerical method allows for a new way to compute steady states of such equations when they exist. We conclude by several numerical experiments to illustrate the sharpness and relevance of our theoretical results, as well as to examine situations that do not meet the hypotheses of our theoretical results, and to illustrate how our results extend to higher dimensions.

Uniform estimates for a fully discrete scheme integrating the linear heat equation on a bounded interval with pure Neumann boundary conditions

TL;DR

The paper addresses uniform-in-time error estimation for a fully discrete scheme solving the linear heat equation with Neumann boundary conditions on a bounded interval. It combines symmetric finite-difference spatial discretization with explicit Euler time stepping under a CFL condition and leverages a spectral analysis of the discrete Laplacian, together with a careful decomposition of boundary-induced consistency defects, to prove a uniform convergence rate of order (and in a Corollary) for the homogeneous problem, extended to the nonhomogeneous case and to steady-state computation. A detailed error analysis partitions the error into propagation and consistency components, analyzes boundary terms via a spectral decomposition, and establishes uniform-in-time bounds through discrete Gronwall-type arguments and exponential decay properties. The work also shows how to compute steady states by time-stepping a nonhomogeneous heat equation and proves corresponding uniform-in-time error bounds, with extensions to higher dimensions demonstrated numerically. Overall, the results provide a robust framework for long-time integration of parabolic problems with Neumann boundary conditions and offer practical tools for steady-state computations in both 1D and 2D settings.

Abstract

This manuscript deals with the analysis of numerical methods for the full discretization (in time and space) of the linear heat equation with Neumann boundary conditions, and it provides the reader with error estimates that are uniform in time. First, we consider the homogeneous equation with homogeneous Neumann boundary conditions over a finite interval. Using finite differences in space and the Euler method in time, we prove that our method is of order 1 in space, uniformly in time, under a classical CFL condition, and despite its lack of consistency at the boundaries. Second, we consider the nonhomogeneous equation with nonhomogeneous Neumann boundary conditions over a finite interval. Using a tailored similar scheme, we prove that our method is also of order 1 in space, uniformly in time, under a classical CFL condition. We indicate how this numerical method allows for a new way to compute steady states of such equations when they exist. We conclude by several numerical experiments to illustrate the sharpness and relevance of our theoretical results, as well as to examine situations that do not meet the hypotheses of our theoretical results, and to illustrate how our results extend to higher dimensions.
Paper Structure (32 sections, 17 theorems, 157 equations, 4 figures)

This paper contains 32 sections, 17 theorems, 157 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setting of the problem and main results
  3. The continuous linear heat equation with homogeneous Neumann boundary conditions
  4. Decay properties of the solutions of \ref{['eq:heat']}
  5. The discretized problem
  6. Description of the lack of consistency
  7. Main results
  8. Error analysis
  9. Spectral analysis of ${\mathsf{P}_\delta}$
  10. Spectral decomposition of ${\mathsf{P}_\delta}$
  11. Decomposition of the boundary terms using the spectral decomposition of ${\mathsf{P}_\delta}$
  12. Estimates of the powers of $(\mathsf{Id}+{\delta t}{\mathsf{P}_\delta})$
  13. Analysis of the term in $\varepsilon_1$
  14. Analysis of the term with $\mathcal{L}_\delta^1$
  15. Analysis of the term with $\mathcal{L}_\delta^2$
  16. ...and 17 more sections

Key Result

Proposition 2.7

Assume $J\geq 2$ and ${\delta t}>0$ are such that the CFL condition eq:CFLheat is fulfilled. We have where $\rho$ denotes the spectral radius.

Figures (4)

  • Figure 1: Numerical error as a function of $J$ for several values of $n{\delta t}$ when approximating the solution of the homogeneous linear heat equation \ref{['eq:heat']} using the scheme \ref{['eq:discrheat']} under the CFL condition \ref{['eq:CFLheat']} (logarithmic scales). The initial datum given in the text : trigonometric polynomial (left panel) and function \ref{['eq:u0pasdansdomP2']} (right panel).
  • Figure 2: Numerical simulation of the solution of the homogeneous linear heat equation \ref{['eq:heat']} associated to the initial datum \ref{['eq:chapeau']} using the scheme \ref{['eq:discrheat']} under the CFL condition \ref{['eq:CFLheat']} for several values of $n{\delta t}$. Left : Numerical approximation $\mathsf{v}^n$ of the solution $u(n{\delta t})$ as a function of $j{\delta \! x}$ computed for several values of $n{\delta t}$ with $J=201$ (multiple scales). Right : Numerical error as a function of $J$ for the same values of $n{\delta t}$ (logarithmic scales).
  • Figure 3: Numerical error as a function of $J$ for several values of $n{\delta t}$ when approximating the solution of the linear nonhomogeneous stationary heat equation \ref{['eq:heat_stat_nh']} using the scheme \ref{['eq:discrheat_nh']} under the CFL condition \ref{['eq:CFLheat']} (logarithmic scales). Initial datum : $\tilde{u}^0=\langle \tilde{u}^\infty\rangle {\mathds 1}+w$ (left panel) and $\tilde{u}^0=\langle \tilde{u}^\infty\rangle {\mathds 1}$ (right panel).
  • Figure 4: On the left panels, numerical solution $\tilde{\mathsf{v}}^n$ obtained at $n{\delta t}=5$ and $J=32$. On the right panels, numerical error $\left\|\mathsf{\Pi}_{{\delta \! x}} \tilde{u}^\infty-\tilde{\mathsf{v}}^n\right\|$ as a function of $J$ for several values of $n{\delta t}$. On the top panels : $(\alpha,x_0,y_0)=(15,1,2)$ so that the derivatives of the solution on the boundary are negligible. On the bottom panels : $(\alpha,x_0,y_0)=(1,0,4)$ so that the derivatives of the solution on the boundary are not negligible.

Theorems & Definitions (42)

  • proof
  • Remark 2.3
  • Remark 2.4
  • proof
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • proof
  • Definition 2.9
  • Definition 2.10
  • ...and 32 more