Numerical methods for nonlocal and nonlinear parabolic equations with applications in hydrology and climatology

Łukasz Płociniczak (Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology)

R 3.28 Tue Z2 13:30-14:10

Many natural and industrial phenomena exhibit nonlocal behaviour in temporal or spatial dimension. The former is responsible for processes for which its whole history influences the present state. The latter, on the other hand, indicates that faraway regions of the domain may have some impact on local points. This is useful in describing media of high heterogeneity.

Partial differential equations that are nonlocal involve one or several integral operators that encode this behaviour. For example, Riemann-Liouville or Caputo derivatives are used in temporal direction, while fractional Laplacian or its relatives describe spatial nonlocality. When it comes to numerical methods the discretization of these requires more care than their classical versions. Moreover, it is usually much more expensive, both on CPU and the memory, to conduct simulations involving nonlocal equations.

In this talk we will present several approaches to discretize nonlocal and nonlinear parabolic equations. These include: transformation into a pure integral equation for the time-fractional porous medium equation and Galerkin spectral methods for a general parabolic equation with temporal nonlocality. We will prove stability and convergence of these methods illustrating all the theoretical results with numerical simulations implemented in Julia programming language with parallelization. The talk is based on [2, 3, 1, 4].

Acknowledgement

Ł.P. has been supported by the National Science Centre, Poland (NCN) under the grant Sonata Bis with a number NCN 2020/38/E/ST1/00153.

References

  1. Hanna Okrasińska-Płociniczak and Łukasz Płociniczak. Second order scheme for self-similar solutions of a time-fractional porous medium equation on the half-line. arXiv preprint arXiv:2106.05138, 2021.

  2. Łukasz Płociniczak. Numerical method for the time-fractional porous medium equation. SIAM journal on numerical analysis, 57(2):638–656, 2019.

  3. Łukasz Płociniczak. Linear galerkin-legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology. arXiv preprint arXiv:2106.05140, 2021.

  4. Łukasz Płociniczak. A linear galerkin numerical method for a strongly nonlinear subdiffusion equation. in preparation, 2021.

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