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.

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.

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 Additive difference scheme for two-dimensional fractional in time diffusion equation

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 217-226 ◽  
Author(s):  
Sandra Hodzic-Ivanovic ◽  
Bosko Jovanovic
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Scheme ◽  
Numerical Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

An additive finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional fractional in time diffusion equation is proposed. Its stability is investigated and a convergence rate estimate is obtained.

scholarly journals Factorized difference scheme for two-dimensional subdiffusion equation in nonhomogeneous media

2016 ◽  
Vol 99 (113) ◽  
pp. 1-13 ◽  
Author(s):  
Aleksandra Delic ◽  
Sandra Hodzic ◽  
Bosko Jovanovic
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Nonhomogeneous Media ◽  
Subdiffusion Equation ◽  
Initial Boundary Value

A factorized finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional subdiffusion equation in nonhomogeneous media is proposed. Its stability and convergence are investigated. The corresponding error bounds are obtained.

scholarly journals Factorized difference scheme for 2D fractional in time diffusion equation

2015 ◽  
Vol 9 (2) ◽  
pp. 199-208 ◽  
Author(s):  
Sandra Hodzic
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Difference Scheme ◽  
Numerical Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

A factorized finite-difference scheme for numerical approximation of initial-boundary value problem for two-dimensional fractional in time diffusion equation is proposed. Its stability is investigated and a convergence rate estimate is obtained.

scholarly journals Modelling Weirs in Two-Dimensional Shallow Water Models

Water ◽  
2021 ◽  
Vol 13 (16) ◽  
pp. 2152
Author(s):  
Gonzalo García-Alén ◽  
Olalla García-Fonte ◽  
Luis Cea ◽  
Luís Pena ◽  
Jerónimo Puertas
Keyword(s):  
Experimental Data ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Model ◽  
Two Dimensional ◽  
Shallow Water Model ◽  
Experimental Dataset ◽  
River Hydraulics ◽  
Shallow Water Models ◽  
Water Models

2D models based on the shallow water equations are widely used in river hydraulics. However, these models can present deficiencies in those cases in which their intrinsic hypotheses are not fulfilled. One of these cases is in the presence of weirs. In this work we present an experimental dataset including 194 experiments in nine different weirs. The experimental data are compared to the numerical results obtained with a 2D shallow water model in order to quantify the discrepancies that exist due to the non-fulfillment of the hydrostatic pressure hypotheses. The experimental dataset presented can be used for the validation of other modelling approaches.

On the convergence of difference schemes for the generalized BBM–Burgers equation

2019 ◽  
Vol 26 (3) ◽  
pp. 341-349 ◽  
Author(s):  
Givi Berikelashvili ◽  
Manana Mirianashvili
Keyword(s):  
Exact Solution ◽  
Sobolev Space ◽  
Difference Scheme ◽  
Absolute Stability ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Algebraic Equations ◽  
The Difference ◽  
Initial Boundary Value

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order {k-1} when the exact solution belongs to the Sobolev space {W_{2}^{k}(Q)} , {1<k\leq 3} .

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