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Strong convergence of an explicit full-discrete scheme for stochastic Burgers-Huxley equation

Yibo Wang, Wanrong Cao, Yanzhao Cao

Abstract

The strong convergence of an explicit full-discrete scheme is investigated for the stochastic Burgers-Huxley equation driven by additive space-time white noise, which possesses both Burgers-type and cubic nonlinearities. To discretize the continuous problem in space, we utilize a spectral Galerkin method. Subsequently, we introduce a nonlinear-tamed exponential integrator scheme, resulting in a fully discrete scheme. Within the framework of semigroup theory, this study provides precise estimations of the Sobolev regularity, $L^\infty$ regularity in space, and Hölder continuity in time for the mild solution, as well as for its semi-discrete and full-discrete approximations. Building upon these results, we establish moment boundedness for the numerical solution and obtain strong convergence rates in both spatial and temporal dimensions. A numerical example is presented to validate the theoretical findings.

Strong convergence of an explicit full-discrete scheme for stochastic Burgers-Huxley equation

Abstract

The strong convergence of an explicit full-discrete scheme is investigated for the stochastic Burgers-Huxley equation driven by additive space-time white noise, which possesses both Burgers-type and cubic nonlinearities. To discretize the continuous problem in space, we utilize a spectral Galerkin method. Subsequently, we introduce a nonlinear-tamed exponential integrator scheme, resulting in a fully discrete scheme. Within the framework of semigroup theory, this study provides precise estimations of the Sobolev regularity, regularity in space, and Hölder continuity in time for the mild solution, as well as for its semi-discrete and full-discrete approximations. Building upon these results, we establish moment boundedness for the numerical solution and obtain strong convergence rates in both spatial and temporal dimensions. A numerical example is presented to validate the theoretical findings.
Paper Structure (8 sections, 12 theorems, 122 equations, 1 table)

This paper contains 8 sections, 12 theorems, 122 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Spatial semi-discretization and error estimates
  4. Fully discrete tamed exponential integrator scheme and error estimates
  5. Estimate of a perturbed equation
  6. Boundedness of moments of the full-discrete approximations
  7. Numerical experiment
  8. Conclusion

Key Result

Lemma 2.1

Under Assumption Asp:nu, for $u, v \in H_{0}^{1} (\mathcal{U})$, it holds that

Theorems & Definitions (21)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 11 more