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

Cheng, Yingda; Christlieb, Andrew; Guo, Wei; Ong, Benjamin W.

doi:10.1007/s10915-016-0328-0

Cited by 11 publications

(8 citation statements)

References 43 publications

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“…Over the past several years, the MOL T methods have been developed for solving the heat equation [4,6,20,24], Maxwell's equations [8], the advection equation and Vlasov equation [9], among others. This methodology can be generalized to solving some nonlinear problems, such as the Cahn-Hilliard equation [4].…”

Section: Introductionmentioning

confidence: 99%

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

Wang

Christlieb

Jiang

et al. 2021

CICP

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In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [11] for convection-diffusion equations, which relies on a special kernel-based formulation of the solutions and successive convolution. However, disadvantages appear when we extend the previous method to our equations, such as inefficient choice of parameters and unprovable stability for high-dimensional problems. To overcome these difficulties, a new kernel-based formulation is designed to approach the spatial derivatives. It maintains the good properties of the original one, including the high order accuracy and unconditionally stable for one-dimensional problems, hence allowing much larger time step evolution compared with other explicit schemes. In additional, without extra computational cost, the proposed scheme can enlarge the available interval of the special parameter in the formulation, leading to less errors and higher efficiency. Moreover, theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well. We present numerical tests for one-and two-dimensional scalar and system, demonstrating the designed high order accuracy and unconditionally stable property of the scheme.

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Section: Introductionmentioning

confidence: 99%

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

Wang

Christlieb

Jiang

et al. 2021

CICP

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“…Moreover, this method is capable of conveniently handling general boundary conditions. This MOL T approach has been applied to solve the heat equation, the Allen-Cahn equation [26] , Maxwell's equations [27] , transport equation and the VP system [23] . Note that the MOL T framework is rarely applied to general nonlinear problems, mainly because efficient algorithms of inverting nonlinear BVPs are lacking.…”

Section: Introductionmentioning

confidence: 99%

A hybrid HWENO-based method of lines transpose approach for Vlasov simulations

Wang¹,

Jiang²,

Zhang³

2021

Journal of University of Science and Technology of China

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A new type hybrid Hermite weighted essentially non-oscillatory ( HWENO) schemes in the implicit method of lines transpose ( MOL T ) framework is designed for solving one-dimensional linear transport equations and further applied to the Vlasov-Poisson (VP) simulations via dimensional splitting. Compared with the WENO-based MOL T method given in J. Comput. Phys. [2016, 327: 337-367], the new proposed hybrid HWENO-based MOL T scheme has two advantages. The first is the HWENO schemes using the stencils narrower than those of the WENO schemes with the same order of accuracy. The second is that the schemes can adapt between the linear scheme and the HWENO scheme automatically. In summary, the hybrid HWENO scheme keeps the simplicity and robustness of the simple WENO scheme, while it has higher efficiency with less numerical errors in smooth regions and less computational costs as well. Benchmark examples are given to demonstrate the robustness and good performance of the proposed scheme.

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“…To approximate the integral equations in BVP, the fast multipole method(FMM) solved the heat, Navier-Stokes and linearized Poisson-Boltmann equation in [16,22], Fourier-continuation alternating-direction(FC-AD) algorithm yields unconditionally stability from O(N 2 ) to O(N log N ) [1,24] and Causley et al [5] reduces the computational complexity of the method from O(N 2 ) to O(N ). A variety of schemes, based on the MOL T formulation, have been developed for solving a range of time-dependent PDEs, including the wave equation [3], the heat equation (e.g., the Allen-Cahn equation [4] and Cahn-Hilliard equation [2]), Maxwell's equations [6], and the Vlasov equation [9].…”

Section: Introductionmentioning

confidence: 99%

A kernel based high order “explicit” unconditionally stable scheme for time dependent Hamilton–Jacobi equations

Christlieb

Guo

Jiang

2019

Journal of Computational Physics

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In [11], the authors developed a class of high-order numerical schemes for the Hamilton-Jacobi (H-J) equations, which are unconditionally stable, yet take the form of an explicit scheme. This paper extends such schemes, so that they are more effective at capturing sharp gradients, especially on nonuniform meshes. In particular, we modify the weighted essentially non-oscillatory (WENO) methodology in the previously developed schemes by incorporating an exponential basis and adapting the previously developed nonlinear filters used to control oscillations. The main advantages of the proposed schemes are their effectiveness and simplicity, since they can be easily implemented on higher-dimensional nonuniform meshes. We perform numerical experiments on a collection of examples, including H-J equations with linear, nonlinear, convex and non-convex Hamiltonians. To demonstrate the flexibility of the proposed schemes, we also include test problems defined on non-trivial geometry.Recent work on the MOL T has involved extending the method to solve more general nonlinear PDEs, for which an integral solution is generally not applicable. This work includes the nonlinear degenerate convection-diffusion equations [10], as well as the H-J equations [11]. The key idea of these papers involved exploiting the linearity of a given differential operator, rather than requiring linearity in the underlying equations. This allowed derivative operators in the problems to be expressed through kernel representations developed for linear problems. Formulating applicable derivative operators in this way ultimately facilitated the stability of the schemes, since a global coupling was introduced through the integral operator. As part of this embedding process, a kernel parameter β was introduced, and through a careful selection, was shown to yield schemes which are A-stable. Remarkably, it was shown that one could couple these representations for the derivative operators with an explicit time-stepping method, such as the strong-stability-preserving Runge-Kutta (SSP-RK) methods [17] and still obtain schemes which maintain unconditional stability [10,11]. To address shock-capturing and control non-physical oscillations, the latter two papers introduced quadrature formulas based on WENO reconstruction, along with a nonlinear filter.This paper seeks to extend the work in [10,11] to the H-J equations (1.1) defined on non-uniformly distributed spatial domains. In particular, several improvements are given. First, we develop the MOL T for mapped grids using a general coordinate transformation function, which allows for a non-uniform distribution of grid points. We show that, with this mapping, our numerical scheme is able to preserve the conservation property for the derivative of the solution to the H-J equation. We also describe a novel WENO-based quadrature for the spatial discretization, which uses a basis consisting of exponential polynomials, to improve the shock capturing capabilities of the method. Another difference in this paper, compared to...

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

Cited by 11 publications

References 43 publications

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

A hybrid HWENO-based method of lines transpose approach for Vlasov simulations

A kernel based high order “explicit” unconditionally stable scheme for time dependent Hamilton–Jacobi equations

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