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.

On Preservation of Positivity in Some Finite Element Methods for the Heat Equation

2015 ◽  
Vol 15 (4) ◽  
pp. 417-437 ◽  
Author(s):  
Panagiotis Chatzipantelidis ◽  
Zoltan Horváth ◽  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Heat Equation ◽  
Piecewise Linear ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dirichlet Boundary Conditions ◽  
Lumped Mass ◽  
The Maximum Principle ◽  
Standard Galerkin Method ◽  
Initial Boundary Value

AbstractWe consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We complement in a number of ways earlier studies of the possible extension of this fact to spatially semidiscrete and fully discrete piecewise linear finite element discretizations, based on the standard Galerkin method, the lumped mass method, and the finite volume element method. We also provide numerical examples that illustrate our findings.

Riemann problems and the WAF method for solving the two-dimensional shallow water equations

1992 ◽  
Vol 338 (1649) ◽  
pp. 43-68 ◽  
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Weighted Average ◽  
Nonlinear Effects ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Riemann Solvers ◽  
Shallow Water Flows ◽  
Initial Boundary Value

An exact Riemann solver for the shallow water equations along with several approximate Riemann solvers are presented. These solutions are then used locally to help compute numerically the global solution of the general initial boundary value problem for the shallow water equations. The numerical method used is the weighted average flux method (WAF) proposed by the author. This is a conservative, shock capturing high resolution TVD method. For shallow water flows where nonlinear effects are important or where abrupt changes (hydraulic jumps) are to be expected the present algorithms can be useful in practice. One and two-dimensional solutions are presented to assess both the Riemann solvers and the WAF method.

A Semi-Implicit Runge–Kutta Time-Difference Scheme for the Two-Dimensional Shallow-Water Equations

2006 ◽  
Vol 134 (10) ◽  
pp. 2916-2926 ◽  
Author(s):  
Sajal K. Kar
Keyword(s):  
Difference Scheme ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Difference ◽  
Water Model ◽  
Two Dimensional ◽  
Runge Kutta ◽  
Initial Boundary Value

Abstract A semi-implicit, two time-level, three-step iterative time-difference scheme is proposed for the two-dimensional nonlinear shallow-water equations in a conservative flux form. After a semi-implicit linearization of the governing equations, the linear gravity wave terms are time discretized implicitly using a second-order trapezoidal scheme applied over each iterative step, whereas the nonlinear terms including horizontal advection and other terms left over from the semi-implicit linearization are time discretized explicitly using a third-order Runge–Kutta scheme. The effectiveness of the scheme in terms of numerical accuracy, stability, and efficiency is established through a forced initial-boundary value problem studied using a two-dimensional shallow-water model.

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.

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.

scholarly journals Method of functional parametrization for solving a semi-periodic initial problem for fourth-order partial differential equations

2020 ◽  
Vol 100 (4) ◽  
pp. 5-16
Author(s):  
A.T. Assanova ◽  
◽  
Zh.S. Tokmurzin ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Periodic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Hyperbolic Type ◽  
Order System ◽  
Partial Differential ◽  
Initial Boundary Value

A semi-periodic initial boundary-value problem for a fourth-order system of partial differential equations is considered. Using the method of functional parametrization, an additional parameter is carried out and the studied problem is reduced to the equivalent semi-periodic problem for a system of integro-differential equations of hyperbolic type second order with functional parameters and integral relations. An interrelation between the semi-periodic problem for the system of integro-differential equations of hyperbolic type and a family of Cauchy problems for a system of ordinary differential equations is established. Algorithms for finding of solutions to an equivalent problem are constructed and their convergence is proved. Sufficient conditions of a unique solvability to the semi-periodic initial boundary value problem for the fourth-order system of partial differential equations are obtained.

Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Lian ◽  
Vicenţiu D. Rădulescu ◽  
Runzhang Xu ◽  
Yanbing Yang ◽  
Nan Zhao
Keyword(s):  
Fourth Order ◽  
Wave Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Point Theory ◽  
Initial Energy ◽  
Damping Term ◽  
Blowup Of Solutions ◽  
Positive Initial Energy ◽  
Initial Boundary Value

Abstract In this paper, we consider the initial boundary value problem for a class of fourth-order wave equations with strong damping term, nonlinear weak damping term, strain term and nonlinear source term in polynomial form. First, the local solution is obtained by using fix point theory. Then, by constructing the potential well structure frame, we get the global existence, asymptotic behavior and blowup of solutions for the subcritical initial energy and critical initial energy respectively. Ultimately, we prove the blowup in finite time of solutions for the arbitrarily positive initial energy case.

scholarly journals Absence of Global Solutions for a Fractional in Time and Space Shallow-Water System

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1127 ◽  
Author(s):  
Mohamed Jleli ◽  
Mokhtar Kirane ◽  
Bessem Samet
Keyword(s):  
Shallow Water ◽  
Water System ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time And Space ◽  
Caputo Fractional Derivatives ◽  
Shallow Water System ◽  
Initial Boundary Value ◽  
Derivatives Of

An initial boundary value problem for a fractional in time and space shallow-water system involving ψ -Caputo fractional derivatives of different orders is considered. Using the test function method, sufficient criteria for the absence of global in time solutions of the system are obtained.

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