Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

Geiser, Jürgen; Hueso, José L.; Martínez, Eulalia

doi:10.3390/math8030302

Cited by 5 publications

(7 citation statements)

References 24 publications

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“…In this work, we also apply the recent theoretical results of our work in [26]. While in that work we took the adaptivity of the splitting results into account, here we consider the application of multi-operator splitting approaches and multi-splitting methods.…”

Section: Serial Iterative Splitting Methodsmentioning

confidence: 99%

“…The serial iterative splitting method is given in the following algorithm. Here, we consider discretization step-size τ = t n+1 − t n , which can also be set adaptively, see [26]. We assume to have time-interval [t n , t n+1 ] with n = 0, 1, .…”

Section: Serial Iterative Splitting Methodsmentioning

confidence: 99%

“…• α = β = 2: viscous Burgers' equation, see [2,26], • α = β = 2 and ν = 0: Diffusion equation, see [2,26], and • α, β ∈ (1, 2): fractional diffusion equation, see [14,27].…”

Section: Introductionmentioning

confidence: 99%

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Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations

2020

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The benefits and properties of iterative splitting methods, which are based on serial versions, have been studied in recent years, this work, we extend the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations. For such interesting partial differential examples with higher dimensional, fractional, and nonlinear terms, we could apply the parallel iterative splitting methods, which allow for accelerating the solver methods and reduce the computational time. Here, we could apply the benefits of the higher accuracy of the iterative splitting methods. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, The flexibilisation with multisplitting methods allows for decomposing large scale operator equations. In combination with iterative splitting methods, which use characteristics of waveform-relaxation (WR) methods, we could embed the relaxation behavior and deal better with the nonlinearities of the operators. We consider the convergence results of the parallel iterative splitting methods, reformulating the underlying methods with a summation of the individual convergence results of the WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods with respect to the synchronous and asynchronous treatments. Furthermore, we present different numerical applications of fluid and phase field problems in order to validate the benefit of the parallel versions.

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Section: Serial Iterative Splitting Methodsmentioning

confidence: 99%

Section: Serial Iterative Splitting Methodsmentioning

confidence: 99%

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Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations

2020

Self Cite

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“…This encourages us to build and study efficient numerical methods for this category of nonlinear PDEs. These include but are not limited to the finite difference method [3], the adaptive finite volume element method [4], the adaptive iterative splitting method [5], and so on. When the diffusion parameter is very small, however, the numerical solutions produced by these approaches are insufficient and show non-physical oscillations.…”

Section: Introductionmentioning

confidence: 99%

An H1-Galerkin Space-Time Mixed Finite Element Method for Semilinear Convection–Diffusion–Reaction Equations

Ren,

He,

2023

Fractal Fract

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In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable q=a(x)ux. An H1-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babusˇka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The L2(L2) norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H1-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function.

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“…The proposed method accuracy and efficiency are researched in this work. Adaptive iterative splitting methods to solve convection-diffusion-reaction problems is proposed in paper [6]. Adaptive techniques are embedded into the splitting approach, when developing this method.…”

Section: Introductionmentioning

confidence: 99%

Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

et al. 2022

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Modeling complex hydrodynamic processes in coastal systems is an important problem of mathematical modeling that cannot be solved analytically. The approximation of convective terms is difficult from the point of view of error reduction. This paper proposes a difference scheme based on a linear combination of the Upwind Leapfrog scheme with 2/3 weight coefficient, and the Standard Leapfrog scheme with 1/3 weight coefficient. The weight coefficients are obtained as a result of solving the problem of minimizing the approximation error. Numerical experiments show the advantage of the developed scheme in comparison with other modifications of the Upwind Leapfrog scheme in the case when the convective transport prevails over the diffusion one. The proposed difference scheme solves transport problems more effectively than classical difference schemes in the case when the Péclet number falls in the range from 2 to 20. It follows that the considered difference scheme allows hydrodynamic problems to be solved in regions of complex shape effectively.

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Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

Cited by 5 publications

References 24 publications

Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations

Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations

An H1-Galerkin Space-Time Mixed Finite Element Method for Semilinear Convection–Diffusion–Reaction Equations

Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

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