scholarly journals Local Discontinuous Galerkin Method for Nonlinear Time-Space Fractional Subdiffusion/Superdiffusion Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Meilan Qiu ◽  
Dewang Li ◽  
Yanyun Wu
Keyword(s):  
Convergence Rate ◽  
Fractional Derivative ◽  
Discontinuous Galerkin ◽  
Numerical Schemes ◽  
Step Size ◽  
Time Step ◽  
Local Discontinuous Galerkin ◽  
Time Space ◽  
Backward Euler ◽  
Heterogeneous Rocks

Fractional partial differential equations with time-space fractional derivatives describe some important physical phenomena. For example, the subdiffusion equation (time order 0<α<1) is more suitable to describe the phenomena of charge carrier transport in amorphous semiconductors, nuclear magnetic resonance (NMR) diffusometry in percolative, Rouse, or reptation dynamics in polymeric systems, the diffusion of a scalar tracer in an array of convection rolls, or the dynamics of a bead in a polymeric network, and so on. However, the superdiffusion case (1<α<2) is more accurate to depict the special domains of rotating flows, collective slip diffusion on solid surfaces, layered velocity fields, Richardson turbulent diffusion, bulk-surface exchange controlled dynamics in porous glasses, the transport in micelle systems and heterogeneous rocks, quantum optics, single molecule spectroscopy, the transport in turbulent plasma, bacterial motion, and even for the flight of an albatross (for more physical applications of fractional sub-super diffusion equations, one can see Metzler and Klafter in 2000). In this work, we establish two fully discrete numerical schemes for solving a class of nonlinear time-space fractional subdiffusion/superdiffusion equations by using backward Euler difference 1<α<2 or second-order central difference 1<α<2/local discontinuous Galerkin finite element mixed method. By introducing the mathematical induction method, we show the concrete analysis for the stability and the convergence rate under the L2 norm of the two LDG schemes. In the end, we adopt several numerical experiments to validate the proposed model and demonstrate the features of the two numerical schemes, such as the optimal convergence rate in space direction is close to Ohk+1. The convergence rate in time direction can arrive at Oτ2−α when the fractional derivative is 0<α<1. If the fractional derivative parameter is 1<α<2 and we choose the relationship as h=C′τ (h denotes the space step size, C′ is a constant, and τ is the time step size), then the time convergence rate can reach to Oτ3−α. The experiment results illustrate that the proposed method is effective in solving nonlinear time-space fractional subdiffusion/superdiffusion equations.

scholarly journals Fully Discrete Local Discontinuous Galerkin Approximation for Time-Space Fractional Subdiffusion/Superdiffusion Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-20
Author(s):  
Meilan Qiu ◽  
Liquan Mei ◽  
Dewang Li
Keyword(s):  
Discontinuous Galerkin ◽  
Galerkin Approximation ◽  
Second Order ◽  
Laplacian Operator ◽  
Numerical Schemes ◽  
Central Difference ◽  
Fully Discrete ◽  
Local Discontinuous Galerkin ◽  
Time Space ◽  
Backward Euler

We focus on developing the finite difference (i.e., backward Euler difference or second-order central difference)/local discontinuous Galerkin finite element mixed method to construct and analyze a kind of efficient, accurate, flexible, numerical schemes for approximately solving time-space fractional subdiffusion/superdiffusion equations. Discretizing the time Caputo fractional derivative by using the backward Euler difference for the derivative parameter (0<α<1) or second-order central difference method for (1<α<2), combined with local discontinuous Galerkin method to approximate the spatial derivative which is defined by a fractional Laplacian operator, two high-accuracy fully discrete local discontinuous Galerkin (LDG) schemes of the time-space fractional subdiffusion/superdiffusion equations are proposed, respectively. Through the mathematical induction method, we show the concrete analysis for the stability and the convergence under theL2norm of the LDG schemes. Several numerical experiments are presented to validate the proposed model and demonstrate the convergence rate of numerical schemes. The numerical experiment results show that the fully discrete local discontinuous Galerkin (LDG) methods are efficient and powerful for solving fractional partial differential equations.

Horizontally Explicit and Vertically Implicit (HEVI) Time Discretization Scheme for a Discontinuous Galerkin Nonhydrostatic Model

2015 ◽  
Vol 143 (3) ◽  
pp. 972-990 ◽  
Author(s):  
Lei Bao ◽  
Robert Klöfkorn ◽  
Ramachandran D. Nair
Keyword(s):  
Discontinuous Galerkin ◽  
Strong Stability ◽  
Euler System ◽  
Second Order ◽  
Spatial Discretization ◽  
Step Size ◽  
Time Step ◽  
Exact Integration ◽  
Time Step Size ◽  
Propagating Waves

Abstract A two-dimensional nonhydrostatic (NH) atmospheric model based on the compressible Euler system has been developed in the (x, z) Cartesian domain. The spatial discretization is based on a nodal discontinuous Galerkin (DG) method with exact integration. The orography is handled by the terrain-following height-based coordinate system. The time integration uses the horizontally explicit and vertically implicit (HEVI) time-splitting scheme, which is introduced to address the stringent restriction on the explicit time step size due to a high aspect ratio between the horizontal (x) and vertical (z) spatial discretization. The HEVI scheme is generally based on the Strang-type operator-split approach, where the horizontally propagating waves in the Euler system are solved explicitly while the vertically propagating waves are treated implicitly. As a consequence, the HEVI scheme relaxes the maximum allowed time step to be mainly determined by the horizontal grid spacing. The accuracy of the HEVI scheme is rigorously compared against that of the explicit strong stability-preserving (SSP) Runge–Kutta (RK) scheme using several NH benchmark test cases. The HEVI scheme shows a second-order temporal convergence, as expected. The results of the HEVI scheme are qualitatively comparable to those of the SSP-RK3 scheme. Moreover, the HEVI DG formulation can also be seamlessly extended to account for the second-order diffusion as in the case of the standard SSP-RK DG formulation. In the presence of orography, the HEVI scheme produces high quality results, which are visually identical to those produced by the SSP-RK3 scheme.

scholarly journals Multiwavelet Discontinuous Galerkin-Accelerated Exact Linear Part (ELP) Method for the Shallow-Water Equations on the Cubed Sphere

2011 ◽  
Vol 139 (2) ◽  
pp. 457-473 ◽  
Author(s):  
Rick Archibald ◽  
Katherine J. Evans ◽  
John Drake ◽  
James B. White
Keyword(s):  
Galerkin Method ◽  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Shallow Water Equations ◽  
Linear Part ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Cubed Sphere

Abstract In this paper a new approach is presented to increase the time-step size for an explicit discontinuous Galerkin numerical method. The attributes of this approach are demonstrated on standard tests for the shallow-water equations on the sphere. The addition of multiwavelets to the discontinuous Galerkin method, which has the benefit of being scalable, flexible, and conservative, provides a hierarchical scale structure that can be exploited to improve computational efficiency in both the spatial and temporal dimensions. This paper explains how combining a multiwavelet discontinuous Galerkin method with exact-linear-part time evolution schemes, which can remain stable for implicit-sized time steps, can help increase the time-step size for shallow-water equations on the sphere.

Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Fengna Yan ◽  
Yan Xu
Keyword(s):  
Error Analysis ◽  
Discontinuous Galerkin ◽  
Implicit Time ◽  
Time Step ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Hilliard Equation ◽  
Local Discontinuous Galerkin ◽  
Energy Stable ◽  
Cahn Hilliard Equation

Abstract In this paper, we mainly study the error analysis of an unconditionally energy stable local discontinuous Galerkin (LDG) scheme for the Cahn–Hilliard equation with concentration-dependent mobility. The time discretization is based on the invariant energy quadratization (IEQ) method. The fully discrete scheme leads to a linear algebraic system to solve at each time step. The main difficulty in the error estimates is the lack of control on some jump terms at cell boundaries in the LDG discretization. Special treatments are needed for the initial condition and the non-constant mobility term of the Cahn–Hilliard equation. For the analysis of the non-constant mobility term, we take full advantage of the semi-implicit time-discrete method and bound some numerical variables in L ∞ L^{\infty} -norm by the mathematical induction method. The optimal error results are obtained for the fully discrete scheme.

A Conservative Local Discontinuous Galerkin Method for the Schrödinger-KdV System

2014 ◽  
Vol 15 (4) ◽  
pp. 1091-1107 ◽  
Author(s):  
Yinhua Xia ◽  
Yan Xu
Keyword(s):  
Discontinuous Galerkin ◽  
Time Discretization ◽  
Local Discontinuous Galerkin Method ◽  
Implicit Time ◽  
Discrete Version ◽  
Time Step ◽  
Local Discontinuous Galerkin ◽  
Ldg Method ◽  
Spatial Derivatives ◽  
Number Of Particles

AbstractIn this paper we develop a conservative local discontinuous Galerkin (LDG) method for the Schrödinger-Korteweg-de Vries (Sch-KdV) system, which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Conservative quantities in the discrete version of the number of plasmons, energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system. Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives. Numerical results for accuracy tests of stationary traveling soliton, and the collision of solitons are shown. Numerical experiments illustrate the accuracy and capability of the method.

scholarly journals Analysis of the Characteristics and Influencing Factors of Gas Explosion in Heading Face

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Xue-bo Zhang ◽  
Jian-liang Gao ◽  
Jing-zhang Ren ◽  
Chun-xia Wang
Keyword(s):  
Numerical Simulation ◽  
Shock Wave ◽  
Influencing Factors ◽  
Geometric Model ◽  
Great Influence ◽  
Ignition Energy ◽  
Gas Explosion ◽  
Step Size ◽  
Time Step ◽  
Time Space

In order to accurately grasp the characteristics and influencing factors of gas explosion in heading face, the mathematical model of gas explosion was determined. According to the actual size of a heading face of a coal mine, a 3D geometric model with a length of 100 m was established, and the effects of ignition energy and gas explosion equivalent on the gas explosion characteristics of the heading face were analyzed. The results show the following. (1) The mathematical models for numerical simulation of gas explosion can accurately simulate the gas explosion and its propagation process. The time-space step size has a great influence on the simulation results. The grid spacing for numerical simulation of mine gas explosion is determined to be 0.1 m and the time step length is determined to be 0.001 s. (2) The ignition energy has a limited effect on gas explosion characteristics. It only has a certain influence on the gas explosion process, but has little influence on the overpressure of shock wave. The larger the ignition energy is, the faster the explosion reaction speed is, and the maximum overpressure increases slightly. When the ignition energy increases to a certain value, the time of peak shock wave and the maximum overpressure both tend to be stable. The ignition energy has little effect on gas explosion characteristics when an explosion accident occurs underground with a large amount of gas accumulation. (3) The gas explosion equivalent has a great influence on the overpressure of gas explosion shock wave. The higher the explosion equivalent is, the greater the pressure is, and the peak value of the shock wave overpressure increases with the explosion equivalent as a power function. The research results have important guiding significance for the research and development of new technology for prevention and control of gas explosion.

scholarly journals New Numerical Aspects of Caputo-Fabrizio Fractional Derivative Operator

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 374 ◽  
Author(s):  
Sania Qureshi ◽  
Norodin A. Rangaig ◽  
Dumitru Baleanu
Keyword(s):  
Fractional Derivative ◽  
Numerical Approximation ◽  
Absolute Difference ◽  
Step Size ◽  
Time Step ◽  
Derivative Operator ◽  
Time Step Size ◽  
First Order ◽  
Forward Difference ◽  
Difference Formula

In this paper, a new definition for the fractional order operator called the Caputo-Fabrizio (CF) fractional derivative operator without singular kernel has been numerically approximated using the two-point finite forward difference formula for the classical first-order derivative of the function f ( t ) appearing inside the integral sign of the definition of the CF operator. Thus, a numerical differentiation formula has been proposed in the present study. The obtained numerical approximation was found to be of first-order convergence, having decreasing absolute errors with respect to a decrease in the time step size h used in the approximations. Such absolute errors are computed as the absolute difference between the results obtained through the proposed numerical approximation and the exact solution. With the aim of improved accuracy, the two-point finite forward difference formula has also been utilized for the continuous temporal mesh. Some mathematical models of varying nature, including a diffusion-wave equation, are numerically solved, whereas the first-order accuracy is not only verified by the error analysis but also experimentally tested by decreasing the time-step size by one order of magnitude, whereupon the proposed numerical approximation also shows a one-order decrease in the magnitude of its absolute errors computed at the final mesh point of the integration interval under consideration.

Fast Solver for the Local Discontinuous Galerkin Discretization of the KdV Type Equations

2015 ◽  
Vol 17 (2) ◽  
pp. 424-457 ◽  
Author(s):  
Ruihan Guo ◽  
Yan Xu
Keyword(s):  
Discontinuous Galerkin ◽  
Linear Equations ◽  
Three Dimensional ◽  
Mode Analysis ◽  
Implicit Time ◽  
Local Mode ◽  
Time Step ◽  
Local Discontinuous Galerkin ◽  
Time Marching ◽  
Kdv Type Equations

AbstractIn this paper, we will develop a fast iterative solver for the system of linear equations arising from the local discontinuous Galerkin (LDG) spatial discretization and additive Runge-Kutta (ARK) time marching method for the KdV type equations. Being implicit in time, the severe time step , with the k-th order of the partial differential equations (PDEs)) restriction for explicit methods will be removed. The equations at the implicit time level are linear and we demonstrate an efficient, practical multigrid (MG) method for solving the equations. In particular, we numerically show the optimal or sub-optimal complexity of the MG solver and a two-level local mode analysis is used to analyze the convergence behavior of the MG method. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency and capability of the LDG method coupled with the multigrid method for solving the KdV type equations.

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