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A Fast Adaptive Method for the Heat Equation with Moving or Free Boundaries in One Dimension

Chengyue Song, Jun Wang

TL;DR

This paper develops a fast, adaptive method for solving the one-dimensional heat equation with moving or free boundaries using an integral-equation formulation. A sum-of-exponentials based continuous fast Gauss transform (FGT) accelerates both the initial heat potential and the double-layer potential, while history dependence is managed by decomposing into a smooth history part and a singular local part on an adaptive volume grid. The authors establish rigorous error bounds for the adaptive history resolution, implement a hybrid quadrature for singular terms, and introduce an efficient bootstrapping time-stepping strategy that yields near-linear time complexity in the number of time steps. They demonstrate the approach through numerical tests on periodic and nonperiodic problems and a Stefan-type moving boundary application, and indicate straightforward generalization to higher dimensions. Overall, the work delivers a robust, parameter-light framework for fast heat-potential evaluations in moving-domain problems with practical impact for simulations involving phase boundaries and nonstationary domains.

Abstract

We present a fast adaptive method for the evaluation of heat potentials, which plays a key role in the integral equation approach for the solution of the heat equation, especially in a non-stationary domain. The algorithm utilizes a sum-of-exponential based fast Gauss transform that evaluates the convolution of a Gaussian with either discrete or continuous volume distributions. The latest implementation of the algorithm allows for both periodic and free space boundary conditions. The history dependence is overcome by splitting the heat potentials into a smooth history part and a singular local part. We discuss the resolution of the history part on an adaptive volume grid in detail, providing sharp estimates that allow for the construction of an optimal grid, justifying the efficiency of the bootstrapping scheme. While the discussion in this paper is restricted to one spatial dimension, the generalization to two and three dimensions is straightforward. The performance of the algorithm is illustrated via several numerical examples.

A Fast Adaptive Method for the Heat Equation with Moving or Free Boundaries in One Dimension

TL;DR

This paper develops a fast, adaptive method for solving the one-dimensional heat equation with moving or free boundaries using an integral-equation formulation. A sum-of-exponentials based continuous fast Gauss transform (FGT) accelerates both the initial heat potential and the double-layer potential, while history dependence is managed by decomposing into a smooth history part and a singular local part on an adaptive volume grid. The authors establish rigorous error bounds for the adaptive history resolution, implement a hybrid quadrature for singular terms, and introduce an efficient bootstrapping time-stepping strategy that yields near-linear time complexity in the number of time steps. They demonstrate the approach through numerical tests on periodic and nonperiodic problems and a Stefan-type moving boundary application, and indicate straightforward generalization to higher dimensions. Overall, the work delivers a robust, parameter-light framework for fast heat-potential evaluations in moving-domain problems with practical impact for simulations involving phase boundaries and nonstationary domains.

Abstract

We present a fast adaptive method for the evaluation of heat potentials, which plays a key role in the integral equation approach for the solution of the heat equation, especially in a non-stationary domain. The algorithm utilizes a sum-of-exponential based fast Gauss transform that evaluates the convolution of a Gaussian with either discrete or continuous volume distributions. The latest implementation of the algorithm allows for both periodic and free space boundary conditions. The history dependence is overcome by splitting the heat potentials into a smooth history part and a singular local part. We discuss the resolution of the history part on an adaptive volume grid in detail, providing sharp estimates that allow for the construction of an optimal grid, justifying the efficiency of the bootstrapping scheme. While the discussion in this paper is restricted to one spatial dimension, the generalization to two and three dimensions is straightforward. The performance of the algorithm is illustrated via several numerical examples.
Paper Structure (23 sections, 77 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 23 sections, 77 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Integral equation formulation of the heat equation in one dimension
  4. SOE approximation
  5. One-dimensional continuous FGT
  6. One-dimensional continuous FGT based on SOE approximation
  7. FGT based on series expansion
  8. periodization
  9. Solving the Volterra Equation
  10. The evaulation of DLHP
  11. Hybrid quadrature method
  12. Time stepping
  13. Construct the spatial mesh
  14. Numerical examples
  15. Numerical results for the FGT
  16. ...and 8 more sections

Figures (4)

  • Figure 1: The computation order of the SOE-based FGT
  • Figure 2: The expression of $\varphi$
  • Figure 3: Schematic of one-dimensional one-phase Stefan problem
  • Figure 4: The figure of solution $s(t)$ of the modified Stefan problem example