scholarly journals Dislocation transport using a time-explicit Runge-Kutta discontinuous Galerkin finite element approach

2021 ◽  
Author(s):  
Manas Vijay Upadhyay ◽  
Jérémy Bleyer
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
Convergence Rates ◽  
Method Of Lines ◽  
Initial Boundary ◽  
Strong Stability ◽  
Initial Boundary Value Problem ◽  
Polynomial Degree ◽  
First Order ◽  
Runge Kutta

Abstract A time-explicit Runge-Kutta discontinuous Galerkin (RKDG) finite element scheme is proposed to solve the dislocation transport initial boundary value problem in 3D. The dislocation density transport equation, which lies at the core of this problem, is a first-order unsteady-state advection-reaction-type hyperbolic partial differential equation; the DG approach is well suited to solve such equations that lack any diffusion terms. The development of the RKDG scheme follows the method of lines approach. First, a space semi-discretization is performed using the DG approach with upwinding to obtain a system of ordinary differential equations in time. Then, time discretization is performed using explicit RK schemes to solve this system. The 3D numerical implementation of the RKDG scheme is performed for the first-order (forward Euler), second-order and third-order RK methods using the strong stability preserving approach. These implementations provide (quasi-)optimal convergence rates for smooth solutions. A slope limiter is used to prevent spurious Gibbs oscillations arising from high-order space approximations (polynomial degree ≥ 1) of rough solutions. A parametric study is performed to understand the influence of key parameters of the RKDG scheme on the stability of the solution predicted during a screw dislocation transport simulation. Then, annihilation of two oppositely signed screw dislocations and the expansion of a polygonal dislocation loop are simulated. The RKDG scheme is able to resolve the shock generated during dislocation annihilation without any spurious oscillations and predict the prismatic loop expansion with very low numerical diffusion. These results demonstrate the robustness of the scheme.

A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems

2013 ◽  
Vol 5 (04) ◽  
pp. 548-568 ◽  
Author(s):  
Tao Lin ◽  
Yanping Lin ◽  
Xu Zhang
Keyword(s):  
Finite Element ◽  
Method Of Lines ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Interface Problem ◽  
Cartesian Mesh ◽  
Moving Interface ◽  
Immersed Finite Elements ◽  
Initial Boundary Value ◽  
Moving Interface Problems

AbstractThis article extends the finite element method of lines to a parabolic initial boundary value problem whose diffusion coefficient is discontinuous across an interface that changes with respect to time. The method presented here uses immersed finite element (IFE) functions for the discretization in spatial variables that can be carried out over a fixed mesh (such as a Cartesian mesh if desired), and this feature makes it possible to reduce the parabolic equation to a system of ordinary differential equations (ODE) through the usual semi-discretization procedure. Therefore, with a suitable choice of the ODE solver, this method can reliably and efficiently solve a parabolic moving interface problem over a fixed structured (Cartesian) mesh. Numerical examples are presented to demonstrate features of this new method.

COMPUTATIONAL ASPECTS OF THE DISCONTINUOUS GALERKIN METHOD FOR THE WAVE EQUATION

2008 ◽  
Vol 16 (04) ◽  
pp. 507-530 ◽  
Author(s):  
TIMO LÄHIVAARA ◽  
MATTI MALINEN ◽  
JARI P. KAIPIO ◽  
TOMI HUTTUNEN
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Wave Propagation ◽  
Wave Equation ◽  
Discontinuous Galerkin ◽  
Homogeneous Medium ◽  
Polynomial Degree ◽  
Dg Method ◽  
Runge Kutta ◽  
Element Method

The Discontinuous Galerkin (DG) method is a powerful tool for numerically simulating wave propagation problems. In this paper, the time-dependent wave equation is solved using the DG method for spatial discretization; and the Crank–Nicolson and fourth-order explicit, singly diagonally implicit Runge–Kutta methods, and, for reference, the explicit Runge–Kutta method, were used for time integration. These simulation methods were studied using two-dimensional numerical experiments. The aim of the experiments was to study the effect of the polynomial degree of the basis functions, grid density, and the Courant–Friedrichs–Lewy number on the accuracy of the approximation. The sensitivity of the methods to distorted finite elements was also examined. Results from the DG method were compared with those computed using a conventional finite element method. Three different model problems were considered. In the first experiment, wave propagation in a homogeneous medium was studied. In the second experiment, the scattering and propagation of waves in an inhomogeneous medium were investigated. The third experiment evaluated wave propagation in a more complicated domain involving multiple scattering waves. The results indicated that the DG method provides more accurate solutions than the conventional finite element method with a reduced computation time and a lower number of degrees of freedom.

Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation

2020 ◽  
Author(s):  
Georgios Akrivis ◽  
Buyang Li
Keyword(s):  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inner Product ◽  
Polynomial Degree ◽  
Finite Element Approximations ◽  
The Cauchy Problem ◽  
Initial Boundary Value

Abstract For a class of compatible profiles of initial data describing bulk phase regions separated by transition zones, we approximate the Cauchy problem of the Allen–Cahn (AC) phase field equation by an initial-boundary value problem in a bounded domain with the Dirichlet boundary condition. The initial-boundary value problem is discretized in time by the backward difference formulae (BDF) of order $1\leqslant q\leqslant 5$ and in space by the Galerkin finite element method of polynomial degree $r-1$, with $r\geqslant 2$. We establish an error estimate of $O(\tau ^q\varepsilon ^{-q-\frac 12}+h^{r}\varepsilon ^{-r-\frac 12}+{e}^{-c/\varepsilon })$ with explicit dependence on the small parameter $\varepsilon$ describing the thickness of the phase transition layer. The analysis utilizes the maximum-norm stability of BDF and finite element methods with respect to the boundary data, the discrete maximal $L^p$-regularity of BDF methods for parabolic equations and the Nevanlinna–Odeh multiplier technique combined with a time-dependent inner product motivated by a spectrum estimate of the linearized AC operator.

A Priori Error Analysis of a Discontinuous Galerkin Scheme for the Magnetic Induction Equation

2020 ◽  
Vol 20 (1) ◽  
pp. 121-140 ◽  
Author(s):  
Tanmay Sarkar
Keyword(s):  
Error Analysis ◽  
Discontinuous Galerkin ◽  
Magnetic Induction ◽  
Time Discretization ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Galerkin Scheme ◽  
Optimal Convergence ◽  
Discontinuous Galerkin Scheme ◽  
Runge Kutta

AbstractWe perform the error analysis of a stabilized discontinuous Galerkin scheme for the initial boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions. In order to obtain the quasi-optimal convergence incorporating second-order Runge–Kutta schemes for time discretization, we need a strengthened {4/3}-CFL condition ({\Delta t\sim h^{4/3}}). To overcome this unusual restriction on the CFL condition, we consider the explicit third-order Runge–Kutta scheme for time discretization. We demonstrate the error estimates in {L^{2}}-sense and obtain quasi-optimal convergence for smooth solution in space and time for piecewise polynomials with any degree {l\geq 1} under the standard CFL condition.

scholarly journals Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1113
Author(s):  
Isaías Alonso-Mallo ◽  
Ana M. Portillo
Keyword(s):  
Numerical Experiments ◽  
Splitting Method ◽  
Method Of Lines ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
High Order ◽  
Time Dependent ◽  
Boundary Values ◽  
Lipschitz Continuous

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

NUMERICAL ANALYSIS OF THE FINITE ELEMENT METHOD APPLIED TO THE BOLTZMANN-POISSON SYSTEM

1995 ◽  
Vol 05 (03) ◽  
pp. 351-365 ◽  
Author(s):  
V. SHUTYAEV ◽  
O. TRUFANOV
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Analysis ◽  
Semiconductor Device ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Poisson System ◽  
The Finite Element Method ◽  
Initial Boundary Value ◽  
Element Method

This paper is concerned with the numerical analysis of the mathematical model for a semiconductor device with the use of the Boltzmann equation. A mixed initial-boundary value problem for nonstationary Boltzmann-Poisson system in the case of one spatial variable is considered. A numerical algorithm for solving this problem is constructed and justified. The algorithm is based on an iterative process and the finite element method. A numerical example is presented.

scholarly journals On the standard Galerkin method with explicit RK4 time stepping for the shallow water equations

2019 ◽  
Vol 40 (4) ◽  
pp. 2415-2449
Author(s):  
D C Antonopoulos ◽  
V A Dougalis ◽  
G Kounadis
Keyword(s):  
Finite Element ◽  
Shallow Water ◽  
Fourth Order ◽  
Shallow Water Equations ◽  
Computational Study ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Courant Number ◽  
Standard Galerkin Method ◽  
Initial Boundary Value

Abstract We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension. We discretize the problem in space by the standard Galerkin finite element method on a quasiuniform mesh and in time by the classical four-stage, fourth order, explicit Runge–Kutta scheme. Assuming smoothness of solutions, a Courant number restriction and certain hypotheses on the finite element spaces, we prove $L^{2}$ error estimates that are of fourth-order accuracy in the temporal variable and of the usual, due to the nonuniform mesh, suboptimal order in space. We also make a computational study of the numerical spatial and temporal orders of convergence, and of the validity of a hypothesis made on the finite element spaces.

scholarly journals NUMERICAL SIMULATION OF MAGNETIC DROPLET DYNAMICS IN A ROTATING FIELD

2013 ◽  
Vol 18 (1) ◽  
pp. 80-96
Author(s):  
Andrejs Cebers ◽  
Harijs Kalis
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Partial Differential Equation ◽  
Method Of Lines ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Continuous Functions ◽  
Rotating Magnetic Field ◽  
Droplet Dynamics ◽  
Ill Posed ◽  
Initial Boundary Value

Dynamics and hysteresis of an elongated droplet under the action of a rotating magnetic field is considered for mathematical modelling. The shape of droplet is found by regularization of the ill-posed initial–boundary value problem for nonlinear partial differential equation (PDE). It is shown that two methods of the regularization – introduction of small viscous bending torques and construction of monotonous continuous functions are equivalent. Their connection with the regularization of the ill-posed reverse problems for the parabolic equation of heat conduction is remarked. Spatial discretization is carried out by the finite difference scheme (FDS). Time evolution of numerical solutions is obtained using method of lines for solving a large system of ordinary differential equations (ODE).

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