Form Of Conservation Laws Research Articles

Solving a model for 1-D, three-phase flow vertical equilibrium processes in a homogeneous porous medium by means of a Weighted Essentially Non Oscillatory numerical scheme

Mathematical models of multi-phase flow are useful in some engineering applications like enhanced oil recovery, filtration of pollutants into subsurface, etc. In this work, we derive a mathematical model for the motion of one-dimensional three-phase flow in a porous medium under the condition of vertical equilibrium, which can be viewed as an extension of some two-phase flow models described in the literature.Our model involves a system of two partial differential equations in the form of viscous conservation laws, whose solutions may contain very sharp transitions. We show that a high-order/high resolution Weighted Essentially Non Oscillatory scheme is an appropriate tool to discretize the buoyancy flux and obtain a well resolved representation of the solution of the model. In addition, we show that the efficiency of the scheme may be improved by using Implicit–Explicit (IMEX) strategies, where the parabolic terms are handled by an implicit discretization.

Hyperbolic submodels of an incompressible viscoelastic Maxwell medium

A two-dimensional motion of an incompressible viscoelastic Maxwell continuum is considered. The system of quasilinear equations describing this motion has both real and complex characteristics. A class of effectively one-dimensionalmotions is analyzed for which the original system of equations is decomposed into a hyperbolic subsystem and a quadrature. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. When one chooses the Jaumann corotational derivative as the invariant derivative, the equations of the submodel remain quasilinear. They can be represented in the form of conservation laws, which allows one to analyze discontinuous solutions to these equations. When one chooses the upper or lower convected derivative, the equations of one-dimensional hyperbolic submodels turn out to be linear. The problem of shear motion between parallel plates and the problem of interaction between the stress field that does not depend on one of the coordinates and a transverse shear flow with initially constant vorticity are studied in detail. It is established that a plane Couette flow in the model with the corotational derivative is unstable in the linear approximation in the class of shear flows if the Weissenberg number is greater than one. The development of small perturbations gives rise to discontinuities in tangential velocities and stresses. The hysteresis phenomenon is observed when the Weissenberg number successively increases and decreases while passing through a critical value. The Couette flow in models with the upper and lower convected derivative remains stable with respect to one-dimensional perturbations.

Divergence-free-preserving high-order schemes for magnetohydrodynamics: An artificial magnetic resistivity method

This paper proposes a new strategy that is very simple, divergence-free, high-order accurate, yet has an effective discontinuous-capturing capability for simulating compressible magnetohydrodynamics (MHD). The new strategy is to construct artificial diffusion terms in a physically-consistent manner, such that the artificial terms act as a diffusion term only in the curl of magnetic field to capture numerical discontinuities in the magnetic field while not affecting the divergence field (thus maintaining divergence-free constraint). The physically-consistent artificial diffusion terms are built into the induction equations in a conservation law form at a partial-differential-equation level. The proposed method may be viewed as adding an artificial magnetic resistivity to the induction equations, and is inherently divergence-free both ideal and resistive MHD, with and without shock waves, and also both inviscid and viscous flows. The method is based on finite difference method with co-located variable arrangement, and we show that any linear finite difference scheme in an arbitrary order of accuracy can be used to discretize the modified governing equations to ensures the divergence-free and the global conservation constraints numerically at the discretization level. The artificial magnetic resistivity is designed to localize automatically in regions of discontinuity in the curl of magnetic field and vanish wherever the flow is sufficiently smooth with respect to the grid scale, thereby maintaining the desirable high-order accuracy of the employed discretization scheme in smooth regions. Two-dimensional smooth and non-smooth ideal MHD problems are considered to validate the capability of the proposed method.

An Optimized Correction Procedure via Reconstruction Formulation for Broadband Wave Computation

AbstractRecently, a new differential discontinuous formulation for conservation laws named the Correction Procedure via Reconstruction (CPR) is developed, which is in-spired by several other discontinuous methods such as the discontinuous Galerkin (DG), the spectral volume (SV)/spectral difference (SD) methods. All of them can be unified under the CPR formulation, which is relatively simple to implement due to its finite-difference-like framework. In this paper, a different discontinuous solution space including both polynomial and Fourier basis functions on each element is employed to compute broad-band waves. Free-parameters introduced in the Fourier bases are optimized to minimize both dispersion and dissipation errors through a wave propagation analysis. The optimization procedure is verified with a mesh resolution analysis. Numerical results are presented to demonstrate the performance of the optimized CPR formulation.

Development of a cell centred upwind finite volume algorithm for a new conservation law formulation in structural dynamics

A novel computational methodology is presented for the numerical analysis of fast transient dynamics phenomena in large deformations. The new mixed formulation can be written in the form of a system of first order conservation laws, where the linear momentum, the deformation gradient tensor and the total energy of the system are used as main conservation variables, leading to identical convergence patterns for both displacements and stresses. A cell centred Finite Volume Method is utilised to carry out the spatial discretisation. Naturally, discontinuity of the conservation variables across control volume interfaces leads to a Riemann problem, whose approximate solution is derived. A suitable numerical interface flux is evaluated by means of the Rankine–Hugoniot jump conditions. We take advantage of the conservative formulation to introduce a Total Variation Diminishing shock capturing technique to improve dramatically the performance of the algorithm in the vicinity of sharp solution gradients. A series of numerical examples will be presented in order to demonstrate the capabilities of the scheme. The new formulation is proven to be very efficient in nearly incompressible and bending dominated scenarios in comparison with classical finite element displacement-based approaches. The proposed numerical framework provides a good balance between accuracy and speed of computation.

Flows in convergent channel: comparison of numerical results of different mathematical models

This study deals with numerical solution of 2D unsteady flow of compressible and incompressible viscous fluid in convergent channel for low inlet airflow velocity. Three mathematical models based on system of Navier-Stokes equations for laminar flow are presented and numerical results of the models in the channel are compared. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel during oscillations. The numerical solution is implemented using the finite volume method and the predictor-corrector MacCormack scheme with Jameson artificial viscosity using a grid of quadrilateral cells. The unsteady grid of quadrilateral cells is considered in the form of conservation laws using Arbitrary Lagrangian-Eulerian method. The numerical results, acquired from a developed program, are presented for inlet velocity \(\hat{u}_{\infty }=4.12\) ms\(^{-1}\) and Reynolds number Re\(_{\infty } = 4 \times 10^3\) and the wall motion frequency 100 Hz.

Modeling of Shallow-Water Equations by Using Implicit Higher-Order Compact Scheme with Application to Dam-Break Problem

The paper deals with the unsteady two-dimensional (2D) non-linear shallow-water equations (SWE) in conservation-law form to capture the fluid flow in transition. Numerical simulations of dam-break flood wave in channel transitions have been performed for inviscid and incompressible flow by using two new implicit higher-order compact (HOC) schemes. The algorithm is second order accurate in time and fourth order accurate in space, on the nine-point stencil using third order non-centered difference at the wall boundaries. To solve the algebraic system, bi-conjugate gradient stabilized method (BiCGStab) with preconditioning has been employed. Although, both the schemes are able to capture both transient and steady state solution of shallow water equations, the scheme expressed in conservative law form is unconditionally stable. The model results have been validated for dam-break problem and compared with the experimental data for dry and wet bed conditions. The model results are found to be in good agreement with the experimental observations. The proposed scheme is useful to solve to capture flow transition with minimal number of nodal points, particularly for hyperbolic system.

Bifurcation of Nonlinear Conservation Laws from the Classical Energy Conservation Law for Internal Gravity Waves in Cylindrical Wave Field

New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave field of an uniformly stratified water affected by the earth’s rotation. The obtained conservation laws are different from the well known energy conservation law for internal waves and they are associated with symmetries of the Boussinesq model. Particularly, it is shown that application of Lie group analysis provide three infinite sets of nontrivial integral conservation laws depending on two arbitrary functions, namely a (t, θ ), b (t, r ) and an arbitrary function c (t, θ, r ) which is given implicitly as a nontrivial solution of a partial differential equation involving a (t, θ ) and b (t, r ).

Numerical simulation and convergence analysis of a finite volume scheme for solving general breakage population balance equations

This paper presents a finite volume scheme (FVS) for solving general breakage population balance equations (BPBEs). In particular, the number density based BPBE is transformed to the form of a mass conservation law. Then it becomes easy to apply the well known FVSs that have an important property of mass conservation. Following Kumar and Warnecke for fixed pivot (FP) method [16] and cell average technique (CAT) [15], the stability and the convergence analysis of the semi-discretized FVS are studied. Unlike the CAT and the FP method, the FVS is second order consistent independently of the type of meshes. We also observe that FVS gives second order convergence rate on four different types of uniform and non-uniform meshes with non-decreasing behavior of mesh width. Nevertheless, one order better accuracy than the FP method is achieved on locally uniform meshes. It is also noticed that on non-uniform random meshes the FVS shows one order and two orders higher accuracy than the CAT and the FP method, respectively. The mathematical results of convergence analysis are validated numerically by taking three test problems. The numerical simulations are also compared with the results obtained by the CAT and the FP method.

A Kinetic Approach in Nonlinear Parabolic Problems with $L^1$-Data

We consider the Cauchy-Dirichlet problem for a nonlinear parabolic equation with L^1 data. We show how the concept of kinetic formulation for conservation laws introduced by P.-L. Lions, B. Perthame and E. Tadmor [A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7 (1994), 169–191]<\i> can be be used to give a new proof of the existence of renormalized solutions. To illustrate this approach, we also extend the method to the case where the equation involves an additional gradient term.

An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations

We present a high-order well-balanced unstructured finite volume (FV) scheme on triangular meshes for modeling weakly nonlinear and weakly dispersive water waves over slowly varying bathymetries, as described by the 2D depth-integrated extended Boussinesq equations of Nwogu, rewritten here in conservation law form. The FV scheme numerically solves the conservative form of the equations following the median dual node-centered approach, for both the advective and dispersive part of the equations. For the advective fluxes, the scheme utilizes an approximate Riemann solver along with a well-balanced topography source term upwinding. Higher order accuracy in space and time is achieved through a MUSCL-type reconstruction technique and through a strong stability preserving explicit Runge–Kutta time stepping. Special attention is given to the accurate numerical treatment of moving wet/dry fronts and boundary conditions. The model is applied to several examples of non-breaking wave propagation over variable topographies and the computed solutions are compared to experimental data. The presented results indicate that the presented FV model is robust and capable of simulating wave transformations from relatively deep to shallow water, providing accurate predictions of the wave's propagation, shoaling and runup.

Invariants and other structural properties of biochemical models as a constraint satisfaction problem.

BackgroundWe present a way to compute the minimal semi-positive invariants of a Petri net representing a biological reaction system, as resolution of a Constraint Satisfaction Problem. The use of Petri nets to manipulate Systems Biology models and make available a variety of tools is quite old, and recently analyses based on invariant computation for biological models have become more and more frequent, for instance in the context of module decomposition.ResultsIn our case, this analysis brings both qualitative and quantitative information on the models, in the form of conservation laws, consistency checking, etc. thanks to finite domain constraint programming. It is noticeable that some of the most recent optimizations of standard invariant computation techniques in Petri nets correspond to well-known techniques in constraint solving, like symmetry-breaking. Moreover, we show that the simple and natural encoding proposed is not only efficient but also flexible enough to encompass sub/sur-invariants, siphons/traps, etc., i.e., other Petri net structural properties that lead to supplementary insight on the dynamics of the biochemical system under study.ConclusionsA simple implementation based on GNU-Prolog's finite domain solver, and including symmetry detection and breaking, was incorporated into the BIOCHAM modelling environment and in the independent tool Nicotine. Some illustrative examples and benchmarks are provided.

Comparative analysis of symmetries for the models of mechanics of nonuniform fluids

The symmetries for the main models of mechanics of binary fluids are calculated by the methods of the theory of continuous groups. The fundamental system of the differential form of the main conservation laws is characterized by the ten-parametric Galilean group. The Navier-Stokes set of equations possesses an extended set of symmetries with infinite dimensionality. Simplification of the model changes the order of the set of equations of motion, which leads to the impossibility to take into account the complete set of boundary conditions and formation of discontinuities in solutions for reduced models.

A conservation law formulation of nonlinear elasticity in general relativity

We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress–energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient ψAi. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state. We discuss the Newtonian limit of our formalism and its relation to a second formalism also used in Newtonian elasticity. We validate our framework by numerically solving a set of Riemann problems in Minkowski spacetime, as well as Newtonian ones from the literature.

Exterior differential systems and prolongations for three important nonlinear partial differential equations

Partial differential systems which haveapplications to water waves will be formulated asexterior differential systems. A prolongation structureis determined for each of the equations. The formalismfor studying prolongations is reviewed and theprolongation equations are solved for each equation.One of thesedifferential systems includes the Camassa-Holm andDegasperis-Procesi equations as special cases.The formulation of conservation laws for eachof the systems introduced is discussedand a single example for each is given.It is shown how a Bäcklund transformationfor the last case can be obtained usingthe prolongation results.

Holographic non-abelian charged hydrodynamics from the dynamics of null horizons

We analyze the dynamics of a four-dimensional null hypersurface in a five-dimensional bulk spacetime with Einstein-Yang-Mills fields. In an appropriate ansatz, the projection of the field equations onto the hypersurface takes the form of conservation laws for relativistic hydrodynamics with global non-abelian charges. A Chern-Simons term in the bulk action corresponds to anomalies in the global charges, with a vorticity term arising in the hydrodynamics. We derive the entropy current and obtain unique expressions for some of the leading-order transport coefficients (in the abelian case, all of them) for arbitrary equations of state. As a special case and a concrete example, we discuss the event horizon of a boosted Einstein-Yang-Mills black brane in an asymptotically Anti-de-Sitter spacetime. The evolution equations in that case describe the hydrodynamic limit of a conformal field theory with anomalous global non-abelian charges on the Anti-de-Sitter boundary.

On the discontinuous Galerkin computation of N‐waves focusing

AbstractSonic boom focusing phenomenon can be predicted using the solution to the nonlinear Tricomi equation which is a hybrid (hyperbolic‐elliptic) second‐order partial differential equation. In this paper, the hyperbolic conservation law form is derived, which is valid in the entire domain. In this manner, the presence of two regions where the equation behaves differently (hyperbolic in the upper and elliptic in the lower half‐plane) is avoided. On the upper boundary, a new mixed boundary condition for the acoustic pressure is employed. The discretization is carried out using a discontinuous Galerkin (DG) method combined with a Runge–Kutta total‐variation diminishing scheme. The results show the accuracy of DG methods to solve problems involving sharp gradients and discontinuities. Comparisons with analytical results for the linear case, and other numerical results using classical explicit and compact finite difference schemes and weighted essentially non‐oscillatory schemes are included. Copyright © 2010 John Wiley & Sons, Ltd.

Moment of momentum equation for curvilinear free-surface flow

The control volume approach in fluid mechanics relies upon the conservation of energy, momentum and angular momentum. However, open channel flows are typically related to only conservation of energy and momentum. Thus, for flows of non-uniform velocity and non-hydrostatic pressure distributions, the number of unknowns is larger than that of equations for depth-averaged models. This gap resulted in semi-empirical approaches either using momentum or using energy conservation. However, a more fundamental depth-averaged model may be developed by accounting for angular momentum balance. This basic approach was largely overlooked in hydraulic practice. The generalized form of the control volume conservation laws in curvilinear flow are presented herein, including the angular momentum balance. The development is used to prove that the classical Boussinesq-type solution does not satisfy the angular momentum balance. A more general Boussinesq approach is considered and applied to the 2D free overfall, indicating its ability to reproduce the salient flow features.

Mass, momentum and energy conservation laws in zero-pressure gas dynamics and δ-shocks: II

We study δ-shocks in a one-dimensional system of zero-pressure gas dynamics. In contrast to well-known papers (see References) this system is considered in the form of mass, momentum and energy conservation laws. In order to define such singular solutions, special integral identities are introduced which extend the concept of classical weak solutions. Using these integral identities, the Rankine–Hugoniot conditions for δ-shocks are obtained. It is proved that the mass, momentum and energy transport processes between the area outside the of one-dimensional δ-shock wave front and this front are going on such that the total mass, momentum and energy are independent of time, while the mass and energy concentration processes onto the moving δ-shock wave front are going on. At the same time the total kinetic energy transforms into total internal energy.

On families of compact fourth- and fifth-order approximations involving the inversion of two-point operators for equations with convective terms

New one-parameter families of compact approximations of the first-order derivatives are presented that use inversion procedures for two-point operators. Properties of these families are investigated, and finite difference schemes based on those families for problems with convective terms written in the form of conservation laws are described. Estimates of the accuracy of numerical solutions to benchmark problems are given.