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Crank-Nicolson schemes for sub-diffusion equations with nonsingular and singular source terms in time

Han Zhou, Wenyi Tian

Abstract

In this work, two Crank-Nicolson schemes without corrections are developed for sub-diffusion equations. First, we propose a Crank-Nicolson scheme without correction for problems with regularity assumptions only on the source term. Second, since the existing Crank-Nicolson schemes have a severe reduction of convergence order for solving sub-diffusion equations with singular source terms in time, we then extend our scheme and propose a new Crank-Nicolson scheme for problems with singular source terms in time. Second-order error estimates for both the two Crank-Nicolson schemes are rigorously established by a Laplace transform technique, which are numerically verified by some numerical examples.

Crank-Nicolson schemes for sub-diffusion equations with nonsingular and singular source terms in time

Abstract

In this work, two Crank-Nicolson schemes without corrections are developed for sub-diffusion equations. First, we propose a Crank-Nicolson scheme without correction for problems with regularity assumptions only on the source term. Second, since the existing Crank-Nicolson schemes have a severe reduction of convergence order for solving sub-diffusion equations with singular source terms in time, we then extend our scheme and propose a new Crank-Nicolson scheme for problems with singular source terms in time. Second-order error estimates for both the two Crank-Nicolson schemes are rigorously established by a Laplace transform technique, which are numerically verified by some numerical examples.
Paper Structure (11 sections, 14 theorems, 101 equations, 6 tables)

This paper contains 11 sections, 14 theorems, 101 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Crank-Nicolson scheme for nonsingular source terms
  4. Crank-Nicolson scheme I
  5. Solution representations
  6. Error estimates for nonsingular source terms
  7. Crank-Nicolson scheme for singular source terms
  8. Crank-Nicolson scheme II
  9. Error estimates for singular source terms
  10. Numerical examples
  11. Conclusions

Key Result

Theorem 2.1

Let $u^{0}(x)\equiv0$ and $f(x,t)$ in eq:tfdesub satisfy Assumption asm:af (in Section sec:4.2). Then the problem eq:tfdesub has a unique solution $u\in C((0, T]; L^{2}(\Omega))$, which satisfies

Theorems & Definitions (28)

  • Theorem 2.1: ZhouT:2022
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Theorem 3.1
  • Lemma 3.6
  • ...and 18 more