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A fitted scheme for a Caputo initial-boundary value problem

J. L. Gracia, E. O'Riordan, M. Stynes

TL;DR

This work introduces a fitted finite-difference scheme on graded temporal meshes for initial-boundary value problems with Caputo time derivatives of order $α\in(0,1)$. By decomposing the continuous solution into a singular leading term and a smoother remainder, and by exploiting a discrete maximum principle and barrier functions, the authors derive sharp truncation-error and stability estimates that yield improved convergence over the standard L1 method, with an optimal mesh-grading exponent $r=\max\{1,(2-α)/(2α)\}$. The resulting error bound in the discrete maximum norm scales as $|u-u_h|\le C\max\{T^{2α},T^{2-α}\}N^{-{\min\{2-α,2rα\}}}+Ch^2$, highlighting the dependence on the final time $T$ and the benefit of reduced mesh grading. Numerical experiments on multiple test problems validate the theoretical results, demonstrating the fitted scheme’s superior accuracy on both uniform and graded meshes and confirming the $T$-dependence predicted by the analysis.

Abstract

In this paper we consider an initial-boundary value problem with a Caputo time derivative of order $α\in(0,1)$. The solution typically exhibits a weak singularity near the initial time and this causes a reduction in the orders of convergence of standard schemes. To deal with this singularity, the solution is computed with a fitted difference scheme on a graded mesh. The convergence of this scheme is analysed using a discrete maximum principle and carefully chosen barrier functions. Sharp error estimates are proved, which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes, and also indicate an optimal choice for the mesh grading. This optimal mesh grading is less severe than the optimal grading for the standard L1 scheme. Furthermore, the dependence of the error on the final time forms part of our error estimate. Numerical experiments are presented which corroborate our theoretical results.

A fitted scheme for a Caputo initial-boundary value problem

TL;DR

This work introduces a fitted finite-difference scheme on graded temporal meshes for initial-boundary value problems with Caputo time derivatives of order . By decomposing the continuous solution into a singular leading term and a smoother remainder, and by exploiting a discrete maximum principle and barrier functions, the authors derive sharp truncation-error and stability estimates that yield improved convergence over the standard L1 method, with an optimal mesh-grading exponent . The resulting error bound in the discrete maximum norm scales as , highlighting the dependence on the final time and the benefit of reduced mesh grading. Numerical experiments on multiple test problems validate the theoretical results, demonstrating the fitted scheme’s superior accuracy on both uniform and graded meshes and confirming the -dependence predicted by the analysis.

Abstract

In this paper we consider an initial-boundary value problem with a Caputo time derivative of order . The solution typically exhibits a weak singularity near the initial time and this causes a reduction in the orders of convergence of standard schemes. To deal with this singularity, the solution is computed with a fitted difference scheme on a graded mesh. The convergence of this scheme is analysed using a discrete maximum principle and carefully chosen barrier functions. Sharp error estimates are proved, which show an enhancement in the convergence rate compared with the standard L1 approximation on uniform meshes, and also indicate an optimal choice for the mesh grading. This optimal mesh grading is less severe than the optimal grading for the standard L1 scheme. Furthermore, the dependence of the error on the final time forms part of our error estimate. Numerical experiments are presented which corroborate our theoretical results.
Paper Structure (8 sections, 9 theorems, 125 equations, 10 tables)

This paper contains 8 sections, 9 theorems, 125 equations, 10 tables.

Table of Contents

  1. Introduction
  2. Decomposition of the continuous solution
  3. The discretisation
  4. Convergence analysis of the difference scheme
  5. Truncation error estimates for the temporal fractional derivative
  6. Stability and error estimates
  7. Numerical results
  8. Numerical examples

Key Result

Lemma 1

Assume that $\phi\in C^4[0,l]$, $c\in C^2[0,l]$ and $f,f_x,f_{xx}\in C(\bar{Q})$. Assume also that $f (0,0)= f (l,0)=0$ and $\phi "(0)= \phi "(l)=0$. Then the solution of the problem prob can be decomposed as where the remainder term $v(x,t)$ is the solution of the following problem:

Theorems & Definitions (27)

  • Lemma 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 17 more