scholarly journals Finite volume schemes for Boussinesq type equations

2020 ◽  
Author(s):  
Denys Dutykh ◽  
Theodoros Katsaounis ◽  
Dimitrios Mitsotakis
Keyword(s):  
Finite Volume ◽  
Space Dimension ◽  
Type Systems ◽  
Approximate Solutions ◽  
Finite Volume Methods ◽  
Nonlinear Phenomena ◽  
Dispersive Wave ◽  
Finite Volume Schemes ◽  
Wave Models ◽  
Nonlinear Dispersive Wave

Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions.

scholarly journals Finite volume schemes for Boussinesq type equations

2020 ◽  
Author(s):  
Denys Dutykh ◽  
Theodoros Katsaounis ◽  
Dimitrios Mitsotakis
Keyword(s):  
Finite Volume ◽  
Space Dimension ◽  
Type Systems ◽  
Approximate Solutions ◽  
Finite Volume Methods ◽  
Nonlinear Phenomena ◽  
Dispersive Wave ◽  
Finite Volume Schemes ◽  
Wave Models ◽  
Nonlinear Dispersive Wave

Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions.

scholarly journals Finite volume schemes for diffusion equations: Introduction to and review of modern methods

2014 ◽  
Vol 24 (08) ◽  
pp. 1575-1619 ◽  
Author(s):  
Jerome Droniou
Keyword(s):  
Finite Volume ◽  
Approximate Solutions ◽  
Finite Volume Methods ◽  
Diffusion Equations ◽  
Maximum Principles ◽  
Finite Volume Schemes ◽  
Real World Applications ◽  
Continuous Equation ◽  
The Stability ◽  
Main Ideas

We present Finite Volume methods for diffusion equations on generic meshes, that received important coverage in the last decade or so. After introducing the main ideas and construction principles of the methods, we review some literature results, focusing on two important properties of schemes (discrete versions of well-known properties of the continuous equation): coercivity and minimum–maximum principles. Coercivity ensures the stability of the method as well as its convergence under assumptions compatible with real-world applications, whereas minimum–maximum principles are crucial in case of strong anisotropy to obtain physically meaningful approximate solutions.

scholarly journals Finite volume methods for unidirectional dispersive wave models

2012 ◽  
Vol 71 (6) ◽  
pp. 717-736 ◽  
Author(s):  
D. Dutykh ◽  
Th. Katsaounis ◽  
D. Mitsotakis
Keyword(s):  
Finite Volume ◽  
Finite Volume Methods ◽  
Dispersive Wave ◽  
Wave Models

Transport Schemes in GungHo

2021 ◽  
Author(s):  
James Kent
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Mixed Finite Element ◽  
Method Of Lines ◽  
Finite Volume Methods ◽  
Kutta Method ◽  
Finite Volume Schemes ◽  
Dynamical Core ◽  
Runge Kutta Method ◽  
The Stability

<p>GungHo is the mixed finite-element dynamical core under development by the Met Office. A key component of the dynamical core is the transport scheme, which advects density, temperature, moisture, and the winds, throughout the atmosphere. Transport in GungHo is performed by finite-volume methods, to ensure conservation of certain quantaties. There are a range of different finite-volume schemes being considered for transport, including the Runge-Kutta/method-of-lines and COSMIC/Lin-Rood schemes. Additional horizontal/vertical splitting approaches are also under consideration, to improve the stability aspects of the model. Here we discuss these transport options and present results from the GungHo framework, featuring both prescribed velocity advection tests and full dry dynamical core tests. </p>

scholarly journals ANALYSIS AND APPROXIMATION OF A SCALAR CONSERVATION LAW WITH A FLUX FUNCTION WITH DISCONTINUOUS COEFFICIENTS

2003 ◽  
Vol 13 (02) ◽  
pp. 221-257 ◽  
Author(s):  
NICOLAS SEGUIN ◽  
JULIEN VOVELLE
Keyword(s):  
Finite Volume ◽  
Conservation Law ◽  
Discontinuous Coefficients ◽  
Finite Volume Methods ◽  
Finite Volume Schemes ◽  
Flux Function ◽  
Scalar Conservation Law ◽  
Line Of Discontinuity ◽  
Scalar Conservation ◽  
Industrial Context

We study here a model of conservative nonlinear conservation law with a flux function with discontinuous coefficients, namely the equation ut + (k(x)u(1 - u))x = 0. It is a particular entropy condition on the line of discontinuity of the coefficient k which ensures the uniqueness of the entropy solution. This condition is discussed and justified. On the other hand, we perform a numerical analysis of the problem. Two finite volume schemes, the Godunov scheme and the VFRoe-ncv scheme, are proposed to simulate the conservation law. They are compared with two finite volume methods classically used in an industrial context. Several tests confirm the good behavior of both new schemes, especially through the discontinuity of permeability k (whereas a loss of accuracy may be detected when industrial methods are performed). Moreover, a modified MUSCL method which accounts for stationary states is introduced.

scholarly journals A finite volume scheme for the solution of a mixed discrete-continuous fragmentation model

2021 ◽  
Author(s):  
Graham Baird ◽  
Endre Suli
Keyword(s):  
Finite Volume ◽  
Numerical Scheme ◽  
Truncation Error ◽  
Approximate Solutions ◽  
Fragmentation Model ◽  
Finite Volume Scheme ◽  
Finite Volume Schemes ◽  
Operator Semigroups ◽  
Volume Scheme ◽  
Fragmentation Equation

This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the equations. The approximate solutions provided by this scheme are first shown to display conservation of mass and preservation of nonnegativity. Then, by utilising a Dunford-Pettis style argument, the sequence of approximate solutions generated is shown, under given restrictions on the model and the mesh, to converge (weakly) in an appropriate L1 space to a weak solution to the problem. By applying the methods and theory of operator semigroups, we are able to show that these weak solutions are unique and necessarily classical (differentiable) solutions, a degree of regularity not generally established when finite volume schemes are applied to such problems. Furthermore, this approach enabled us to derive a bound for the error induced by the truncation of the mass domain, and also establish the convergence of the truncated solutions as the truncation point is increased without bound. Finally, numerical simulations are performed to investigate the performance of the scheme and assess its rate of convergence as the mesh is refined, whilst also verifying the bound on the truncation error.

scholarly journals On the approximate solutions of fragmentation equations

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170541 ◽  
Author(s):  
Jitraj Saha ◽  
Jitendra Kumar ◽  
Stefan Heinrich
Keyword(s):  
Finite Volume ◽  
Approximate Solutions ◽  
Finite Volume Methods ◽  
Test Problems ◽  
Finite Volume Scheme ◽  
Simple Extension ◽  
Two Dimensional ◽  
One Dimensional ◽  
Proposed Model ◽  
Link Model

A numerical model based on the finite volume scheme is proposed to approximate the binary breakage problems. Initially, it is considered that the particle fragments are characterized by a single property, i.e. particle’s volume. We then investigate the extension of the proposed model for solving breakage problems considering two properties of particles. The efficiency to estimate the different moments with good accuracy and simple extension for multi-variable problems are the key features of the proposed method. Moreover, the mathematical convergence analysis is performed for one-dimensional problems. All mathematical findings and numerical results are validated over several test problems. For numerical validation, we propose the extension of Bourgade & Filbet (2008 Math. Comput. 77 , 851–882. ( doi:10.1090/S0025-5718-07-02054-6 )) model for solving two-dimensional pure breakage problems. In this aspect, numerical treatment of the two-dimensional binary breakage models using finite volume methods can be treated to be the first instance in the literature.

scholarly journals DISCRETE DUALITY FINITE VOLUME SCHEMES FOR DOUBLY NONLINEAR DEGENERATE HYPERBOLIC-PARABOLIC EQUATIONS

2010 ◽  
Vol 07 (01) ◽  
pp. 1-67 ◽  
Author(s):  
B. ANDREIANOV ◽  
M. BENDAHMANE ◽  
K. H. KARLSEN
Keyword(s):  
Finite Volume ◽  
Parabolic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Approximate Solutions ◽  
Dirichlet Boundary Conditions ◽  
Finite Volume Schemes ◽  
Doubly Nonlinear ◽  
Dissipation Inequalities ◽  
Nonlinear Degenerate

We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [43]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around basic a priori estimates, the discrete duality features, Minty–Browder type arguments, and "hyperbolic" L∞weak-⋆ compactness arguments (i.e. propagation of compactness along the lines of Tartar, DiPerna, …). Our results cover the case of non-Lipschitz nonlinearities.

Mathematical Development and Verification of a Finite Volume Model for Morphodynamic Flow Applications

2011 ◽  
Vol 3 (4) ◽  
pp. 470-492 ◽  
Author(s):  
Fayssal Benkhaldoun ◽  
Mohammed Seaïd ◽  
Slah Sahmim
Keyword(s):  
Finite Volume ◽  
Space Dimension ◽  
Jacobian Matrix ◽  
Finite Volume Methods ◽  
Governing Equations ◽  
Hydraulic Flow ◽  
System A ◽  
Long Time ◽  
Flow And Sediment Transport ◽  
Numerical Fluxes

AbstractThe accuracy and efficiency of a class of finite volume methods are investigated for numerical solution of morphodynamic problems in one space dimension. The governing equations consist of two components, namely a hydraulic part described by the shallow water equations and a sediment part described by the Exner equation. Based on different formulations of the morphodynamic equations, we propose a family of three finite volume methods. The numerical fluxes are reconstructed using a modified Roe’s scheme that incorporates, in its reconstruction, the sign of the Jacobian matrix in the morphodynamic system. A well-balanced discretization is used for the treatment of the source terms. The method is well-balanced, non-oscillatory and suitable for both slow and rapid interactions between hydraulic flow and sediment transport. The obtained results for several morphodynamic problems are considered to be representative, and might be helpful for a fair rating of finite volume solution schemes, particularly in long time computations.

Sign in / Sign up

Export Citation Format

Share Document