Theoretical analysis of the Exponential Transversal Method of Lines for the diffusion equation

Salazar, Abraham J.; Raydan, Marcos; Campo, Antônio

doi:10.1002/(sici)1098-2426(200001)16:1<30::aid-num3>3.3.co;2-m

Cited by 8 publications

(17 citation statements)

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“…for a < x < b. We justify the neglect of the homogeneous solution in (20) because, as α → ∞, both exponentials converge to zero for any x = a, b. Showing that this limit holds for the semi-discrete solution (i.e.…”

Section: Consistency Of the Particular Solutionmentioning

confidence: 93%

“…Our reasoning is as follows. If from the update equation (20) we solve for the finite difference approximation of 1 c 2 u tt and neglect the homogeneous solution, we find…”

Section: Consistency Of the Particular Solutionmentioning

confidence: 99%

“…Although the MOL T approach is more commonly used for parabolic problems [20,15], it has been considered sparsely for the wave equation [21]. The work we present here is closely related to an independent set of works recently published [5,19], in which the wave equation is solved using a Fourier continuation-ADI (FC-ADI) method.…”

Section: Introductionmentioning

confidence: 99%

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Method of lines transpose: An implicit solution to the wave equation

et al. 2014

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As a follow up to [6], we provide a detailed description of the numerical implementation of an O(N ), A-stable, second order accurate solution of the wave equation, constructed from semi-discrete boundary value problems. We improve on the previous algorithm by replacing the Lax-type correction used in [6], which was necessary for convergence when ∆t < ∆x/c, with a more accurate spatial quadrature, which we prove is convergent.We also demonstrate that the resulting solver remains fast even in the case of unstructured meshes, can incorporate domain decomposition, and allows for the implementation of Dirichlet, Neumann, periodic and outflow boundary conditions.Building upon results for the 1d formulation, we utilize alternate direction implicit (ADI) splitting to achieve a fast O(N ) solver in higher spatial dimensions. Our solver is built upon line objects and, combined with the flexibility of the integral solver, allows us to solve problems on arbitrary spatial domains, by embedding the boundary in a regular Cartesian mesh. Our solver is designed to couple with particle codes, where scale separation is an issue. We therefore demonstrate the ability of our solver to take time steps well beyond that of the Courant-Friedrichs-Lewy (CFL) stability limit of explicit codes.

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Section: Consistency Of the Particular Solutionmentioning

confidence: 93%

“…Our reasoning is as follows. If from the update equation (20) we solve for the finite difference approximation of 1 c 2 u tt and neglect the homogeneous solution, we find…”

Section: Consistency Of the Particular Solutionmentioning

confidence: 99%

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Method of lines transpose: An implicit solution to the wave equation

et al. 2014

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“…Also note that many of these existing methods are in the Method of Line (MOL) framework, meaning that the spatial variable is first discretized, then the numerical solution is updated in time by coupling a suitable time integrator. In this paper, we consider an alternative approach to advance the solution, which is called the Method of Line Transpose (MOL T ) or Rothes's method in the literature [35,36]. As its name implies, such an approach is defined in an orthogonal fashion of the MOL schemes, i.e., the discretization is first carried out for temporal variable, resulting in a boundary value problem (BVP) at the discrete time levels.…”

Section: Introductionmentioning

confidence: 99%

A WENO-based Method of Lines Transpose approach for Vlasov simulations

Christlieb

Guo

Jiang

2016

Journal of Computational Physics

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In this paper, a high order implicit Method of Line Transpose (MOL T ) method based on a weighted essentially non-oscillatory (WENO) methodology is developed for one-dimensional linear transport equations and further applied to the Vlasov-Poisson (VP) simulations via dimensional splitting. In the MOL T framework, the time variable is first discretized by a diagonally implicit strong-stability-preserving Runge-Kutta method, resulting in a boundary value problem (BVP) at the discrete time levels. Then an integral formulation coupled with a high order WENO methodology is employed to solve the BVP. As a result, the proposed scheme is high order accurate in both space and time and free of oscillations even though the solution is discontinuous or has sharp gradients. Moreover, the scheme is able to take larger time step evolution compared with an explicit MOL WENO scheme with the same order of accuracy. The desired positivity-preserving (PP) property of the scheme is further attained by incorporating a newly proposed high order PP limiter. We perform numerical experiments on several benchmarks including linear advection, solid body rotation problem; and on the Landau damping, two-stream instabilities, bump-on-tail, and plasma sheath by solving the VP system. The efficacy and efficiency of the proposed scheme is numerically verified.

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“…The approach we use in this paper consists of several important components, including the development and the extension of the Method of Line Transpose (MOL T ) framework, the investigation of the scalar and vector potential formulations of Maxwell's equations and their asymptotic limit as key steps to ensure the solver to capture the correct Darwin limit, and an efficient treecode algorithm to further accelerate the computation. The MOL T method we consider in this paper is also known as transverse MOL, and Rothe's method in the literature [35,37]. As the name implies, discretization is carried out in an orthogonal fashion, where the time variable is first discretized, followed by solving the resulting boundary value problems (BVPs) at discrete time levels.…”

Section: Introductionmentioning

confidence: 99%

An Asymptotic Preserving Maxwell Solver Resulting in the Darwin Limit of Electrodynamics

et al. 2016

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In plasma simulations, where the speed of light divided by a characteristic length is at a much higher frequency than other relevant parameters in the underlying system, such as the plasma frequency, implicit methods begin to play an important role in generating efficient solutions in these multi-scale problems. Under conditions of scale separation, one can rescale Maxwell's equations in such a way as to give a magneto static limit known as the Darwin approximation of electromagnetics.In this work, we present a new approach to solve Maxwell's equations based on a Method of Lines Transpose (MOL T ) formulation, combined with a fast summation method with computational complexity O(N log N ), where N is the number of grid points (particles). Under appropriate scaling, we show that the proposed schemes result in asymptotic preserving methods that can recover the Darwin limit of electrodynamics.

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Theoretical analysis of the Exponential Transversal Method of Lines for the diffusion equation

Cited by 8 publications

References 0 publications

Method of lines transpose: An implicit solution to the wave equation

Method of lines transpose: An implicit solution to the wave equation

A WENO-based Method of Lines Transpose approach for Vlasov simulations

An Asymptotic Preserving Maxwell Solver Resulting in the Darwin Limit of Electrodynamics

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