A Comparison of Accuracy and Efficiency of Parallel Solvers for Fractional Power Diffusion Problems

Čiegis, Raimondas; Starikovičius, Vadimas; Margenov, Svetozar; Kriauzienė, Rima

doi:10.1007/978-3-319-78024-5_8

Cited by 4 publications

(7 citation statements)

References 9 publications

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“…In this work, we restrict to uniform space grids to compare the accuracy of the selected numerical methods. Finally, we note that in our previous paper, we have used a smooth function f on the right‐hand side, namely, the 3D super‐Gaussian function, and therefore, no reduction of convergence rate has been observed.…”

Section: Problem Formulationmentioning

confidence: 94%

“…In the literature, several different numerical quadrature formulas are proposed to approximate integral . In our previous studies, we have used the quadrature formula based on a β ‐dependent graded mesh. However, initial computational experiments solving the current 3D test problem have shown superior results using another quadrature formula with uniformly distributed quadrature points y j = kj :

\begin{matrix} alignleft \\ align-1 U_{h}^{M 3} = \frac{2 k \sin (π β)}{π} \sum_{j = - m_{1}}^{m_{2}} e^{2 (β - 1) y_{j}} {(e^{- 2 y_{j}} I_{h} + L_{h})}^{- 1} f_{h}, & align-2 \end{matrix}

where m 1 = ⌈ π 2 /(4 βk 2 )⌉ and m 2 = ⌈ π 2 /(4(1 − β ) k 2 )⌉.…”

Section: Quadrature Methods (M3)mentioning

confidence: 99%

“…In our previous studies, we have used the uniform time‐stepping. However, our initial experiments in solving the current test problem have not shown the second order of convergence in time, even for quite small time steps.…”

Section: Pseudo‐parabolic Methods (M2)mentioning

confidence: 99%

“…In our most recent work, we have started the analysis of parallel algorithms for three‐dimensional (3D) problems. Based on the previous research results, the method M1 was excluded from the list of considered methods as less suitable for 3D case due to high memory requirements.…”

Section: Introductionmentioning

confidence: 99%

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Scalability analysis of different parallel solvers for 3D fractional power diffusion problems

Čiegis

Starikovičius

Margenov

et al. 2019

Concurrency and Computation

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Summary In this paper, we develop and investigate the parallel numerical algorithms for three different state‐of‐the‐art numerical methods for solving the non‐local problems described by fractional powers of elliptic operators. These methods transform the non‐local problem into some local differential problems of elliptic or parabolic type. A two‐level parallelization approach is applied to construct the efficient parallel algorithms using the domain decomposition and master‐slave methods, to deal with the increase in computational complexity. We show and compare the serial and parallel solution times that are required to achieve similar accuracy of the solution using different algorithms. Results of extensive convergence tests are presented solving a three‐dimensional test problem with known decrease of the solution's convergence rate depending on the fractional power coefficient. We analyze and discuss the non‐trivial question, which parallel algorithm is recommended to achieve certain accuracy for the given fractional power coefficient.

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Section: Problem Formulationmentioning

confidence: 94%

\begin{matrix} alignleft \\ align-1 U_{h}^{M 3} = \frac{2 k \sin (π β)}{π} \sum_{j = - m_{1}}^{m_{2}} e^{2 (β - 1) y_{j}} {(e^{- 2 y_{j}} I_{h} + L_{h})}^{- 1} f_{h}, & align-2 \end{matrix}

where m 1 = ⌈ π 2 /(4 βk 2 )⌉ and m 2 = ⌈ π 2 /(4(1 − β ) k 2 )⌉.…”

Section: Quadrature Methods (M3)mentioning

confidence: 99%

Section: Pseudo‐parabolic Methods (M2)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Scalability analysis of different parallel solvers for 3D fractional power diffusion problems

Čiegis

Starikovičius

Margenov

et al. 2019

Concurrency and Computation

Self Cite

View full text Add to dashboard Cite

show abstract

“…al. [17]. The present article differs from these in that there is a focus on recent advances in boundary value problems with nonzero boundary conditions and the inhomogeneous fractional operators that such problems entail.…”

Section: Domainmentioning

confidence: 96%