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Register a new userNumerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation
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In this paper, a fully discrete local discontinuous Galerkin (LDG) finite element method is considered for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. We prove that our scheme is unconditional stable and $L^2$ error estimate for the linear case with the convergence rate $O(h^{k+1}+(Delta t)^2+(Delta t)^frac{alpha}{2}h^{k+1/2})$. Numerical examples are presented to show the efficiency and accuracy of our scheme.
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The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high order fully discrete local discontinuous Galerkin (LDG) method based on the generalized alternating numerical fluxes for the tempered fractional diffusion equation. From a practical point of view, the generalized alternating numerical flux which is different from the purely alternating numerical flux has a broader range of applications. We first design an efficient finite difference scheme to approximate the tempered fractional derivatives and then a fully discrete LDG method for the tempered fractional diffusion equation. We prove that the scheme is unconditionally stable and convergent with the order $O(h^{k+1}+tau^{2-alpha})$, where $h, tau$ and $k$ are the step size in space, time and the degree of piecewise polynomials, respectively. Finally numerical experimets are performed to show the effectiveness and testify the accuracy of the method.
In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian $(-Delta)^{s}$ with $sin(0,1)$ in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than $mathcal{O}(h^{k+frac{1}{2}})$.
We present a new class of iterative schemes for solving initial value problems (IVP) based on discontinuous Galerkin (DG) methods. Starting from the weak DG formulation of an IVP, we derive a new iterative method based on a preconditioned Picard iteration. Using this approach, we can systematically construct explicit, implicit and semi-implicit schemes with arbitrary order of accuracy. We also show that the same schemes can be constructed by solving a series of correction equations based on the DG weak formulation. The accuracy of the schemes is proven to be $min{2p+1, K+1}$ with $p$ the degree of the DG polynomial basis and $K$ the number of iterations. The stability is explored numerically; we show that the implicit schemes are $A$-stable at least for $0 leq p leq 9$. Furthermore, we combine the methods with a multilevel strategy to accelerate their convergence speed. The new multilevel scheme is intended to provide a flexible framework for high order space-time discretizations and to be coupled with space-time multigrid techniques for solving partial differential equations (PDEs). We present numerical examples for ODEs and PDEs to analyze the performance of the new methods. Moreover, the newly proposed class of methods, due to its structure, is also a competitive and promising candidate for parallel in time algorithms such as Parareal, PFASST, multigrid in time, etc.
We propose a weak Galerkin(WG) finite element method for solving the one-dimensional Burgers equation. Based on a new weak variational form, both semi-discrete and fully-discrete WG finite element schemes are established and analyzed. We prove the existence of the discrete solution and derive the optimal order error estimates in the discrete $H^1$-norm and $L^2$-norm, respectively. Numerical experiments are presented to illustrate our theoretical analysis.
The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in science and engineering. It is an integro-differential equation involving time, space and angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needs to handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions. Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinate method for angular discretization. The former employs a dynamic mesh adaptation strategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency are demonstrated in a selection of one- and two-dimensional numerical examples.
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