A Comparison of Parallel Algorithms for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

Čiegis, Raimondas; Dapšys, Ignas; Čiegis, Remigijus

doi:10.3390/axioms11030098

Cited by 3 publications

(9 citation statements)

References 23 publications

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“…As an example, we applied this criterion for the scheme (16) and the problem (7). The results of the computational experiments confirm that the condition R 1 is satisfied for fractional power parameters α = 0.25, 0.5, 0.75, space meshes with J = (J x + 1) 3 , J x = 200, 400, and time steps τ = 10 −k , k = 1, .…”

Section: Discrete Schemes Based On Rational Approximationsmentioning

confidence: 63%

“…This paper is a continuation of our previous work [7]. Recently, much attention has been devoted to the solution of multi-dimensional parabolic problems with fractional power elliptic operators.…”

Section: Introductionmentioning

confidence: 78%

“…Such a delicate interlacing of space and time approximation errors is nicely seen from the experimental results. We solve the differential problem (7) of Example 1. Table 1 gives, for a sequence of decreasing time step widths τ, the errors E(τ), and the experimental convergence rates ρ(τ) of the discrete solution for the BE scheme (10) in the maximum norm:…”

Section: Backward Euler Methodsmentioning

confidence: 99%

“…Example 1. In this example, we solve a linear fractional heat equation with a linear source term in three-dimensional space Ω = [0, 1] × [0, 1] × [0, 1] (see [7]):…”

Section: Let Us Solve the Standard Eigen-problemmentioning

confidence: 99%

“…As expected, the accuracy of the approximation is improved in comparison with the results presented in Table 3. Next, we solve the differential problem (7) of Example 1. Table 5 gives for a sequence of decreasing time step widths τ the errors E m (τ) of the discrete solution for the BURA-BRASIL scheme (16) in the maximum norm:…”

Section: Discrete Schemes Based On Rational Approximationsmentioning

confidence: 99%

See 4 more Smart Citations

On a Framework for the Stability and Convergence Analysis of Discrete Schemes for Nonstationary Nonlocal Problems of Parabolic Type

Čiegis

Dapšys

2022

Mathematics

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The main aim of this article is to propose a general framework for the theoretical analysis of discrete schemes used to solve multi-dimensional parabolic problems with fractional power elliptic operators. This analysis is split into three parts. The first part is based on techniques well developed for the solution of nonlocal elliptic problems. The obtained discrete elliptic operators are used to formulate semi-discrete approximations. Next, the fully discrete schemes are constructed by applying the classical and robust approximations of time derivatives. The existing stability and convergence results are directly included in the new framework. In the third part, approximations of transfer operators are constructed by using uniform and the best uniform rational approximations. The stability and accuracy of the obtained local discrete schemes are investigated. The results of computational experiments are presented and analyzed. A three-dimensional test problem is solved. The rational approximations are constructed by using the BRASIL algorithm.

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Section: Discrete Schemes Based On Rational Approximationsmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 78%

Section: Backward Euler Methodsmentioning

confidence: 99%

“…Example 1. In this example, we solve a linear fractional heat equation with a linear source term in three-dimensional space Ω = [0, 1] × [0, 1] × [0, 1] (see [7]):…”

Section: Let Us Solve the Standard Eigen-problemmentioning

confidence: 99%

Section: Discrete Schemes Based On Rational Approximationsmentioning

confidence: 99%

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On a Framework for the Stability and Convergence Analysis of Discrete Schemes for Nonstationary Nonlocal Problems of Parabolic Type

Čiegis

Dapšys

2022

Mathematics

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On Construction of Partially Dimension-Reduced Approximations for Nonstationary Nonlocal Problems of a Parabolic Type

et al. 2023

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The main aim of this article is to propose an adaptive method to solve multidimensional parabolic problems with fractional power elliptic operators. The adaptivity technique is based on a very efficient method when the multidimensional problem is approximated by a partially dimension-reduced mathematical model. Then in the greater part of the domain, only one-dimensional problems are solved. For the first time such a technique is applied for problems with nonlocal diffusion operators. It is well known that, even for classical local diffusion operators, the averaged flux conjugation conditions become nonlocal. Efficient finite volume type discrete schemes are constructed and analysed. The stability and accuracy of obtained local discrete schemes is investigated. The results of computational experiments are presented and compared with theoretical results.

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Numerical Simulation of Fractional Power Diffusion Biosensors

Dapšys

Čiegis

2023

Mathematical Modelling and Analysis

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The main aim of this paper is to propose new mathematical models for simulation of biosensors and to construct and investigate discrete methods for the efficient solution of the obtained systems of nonlinear PDEs. The classical linear diffusion operators are substituted with nonlocal fractional powers of elliptic operators. The splitting type finite volume scheme is used as a basic template for the introduction of new mathematical models. Therefore the accuracy of the splitting scheme is investigated and compared with the symmetric Crank-Nicolson scheme. The dependence of the approximation error on the regularity of the solution is investigated. Results of computational experiments for different values of fractional parameters are presented and analysed.

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A Comparison of Parallel Algorithms for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

Cited by 3 publications

References 23 publications

On a Framework for the Stability and Convergence Analysis of Discrete Schemes for Nonstationary Nonlocal Problems of Parabolic Type

On a Framework for the Stability and Convergence Analysis of Discrete Schemes for Nonstationary Nonlocal Problems of Parabolic Type

On Construction of Partially Dimension-Reduced Approximations for Nonstationary Nonlocal Problems of a Parabolic Type

Numerical Simulation of Fractional Power Diffusion Biosensors

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