Path-conservative Scheme Research Articles

An Entropy Stable Discontinuous Galerkin Method for the Two-Layer Shallow Water Equations on Curvilinear Meshes

We present an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes. We mimic the continuous entropy analysis on the semi-discrete level with the DGSEM constructed on Legendre–Gauss–Lobatto (LGL) nodes. The use of LGL nodes endows the collocated nodal DGSEM with the summation-by-parts property that is key in the discrete analysis. The approximation exploits an equivalent flux differencing formulation for the volume contributions, which generate an entropy conservative split-form of the governing equations. A specific combination of a numerical surface flux and discretization of the nonconservative terms is then applied to obtain a high-order path-conservative scheme that is entropy conservative. Furthermore, we find that this combination yields an analogous discretization for the pressure and nonconservative terms such that the numerical method is well-balanced for discontinuous bathymetry on curvilinear domains. Dissipation is added at the interfaces to create an entropy stable approximation that satisfies the second law of thermodynamics in the discrete case, while maintaining the well-balanced property. We conclude with verification of the theoretical findings through numerical tests and demonstrate results about convergence, entropy stability and well-balancedness of the scheme.

The exact Riemann Solver to the Shallow Water equations for natural channels with bottom steps

The Shallow Water equations model is commonly used to reproduce a wide variety of environmental flows. Its applications include river hydraulics and flood forecasting.Therefore, an algorithm to solve the Riemann problem for the Shallow Water equations can be a considerable asset for the development of numerical schemes devoted to the solution of such equations. Nevertheless, for real world applications, it becomes mandatory to consider the influence of the irregular shape of the cross section of natural channels in the solution.This work presents an energy based exact Riemann solver for the Shallow Water equations for channels with an arbitrarily shaped cross section with the presence of a non flat bottom and dry states.The procedure for solving the Riemann problem involves the solution of a system of non linear algebraic equations, for which a CPU time efficient solution strategy based on nested Newton-Raphson methods is presented.The proposed solver can be used to develop Finite Volume or Discontinuous Galerkin schemes, based on the Godunov method. For a non flat bottom, the Shallow Water equations are a non conservative system, once the bottom elevation is included in the state variables. Many available schemes take advantage of this fact to use approximate path-conservative Riemann solvers. In other cases, the term related to the non flat bottom is treated as a source term. The use of this exact solver avoids the necessity of considering the bottom topography as a source term and of reverting to approximate solvers. These strategies can in facts pose numerical problems in situations such as transonic rarefactions and flows approaching abrupt changes in bottom topography.The solver is tested by reproducing the solution of Riemann problems with a Godunov finite volume scheme. An analysis of the performance of the solver shows that it requires less computational time than path-conservative schemes and that it has better stability and convergence properties, especially in situations involving resonant waves, such as transonic flows across a bottom step.

A splitting scheme for the coupled Saint-Venant-Exner model

We present a splitting method for the one-dimensional Saint-Venant-Exner equations used for describing the bed evolution in shallow water systems. We adapt the flux vector splitting approach of Toro and Vázquez-Cendón (2012) and identify one subsystem of conservative equations (advection system) and one of non-conservative equations (pressure system), both having a very simple eigenstructure compared to the full system. The final numerical scheme is constructed using a Godunov-type path-conservative scheme for the pressure system and a simple conservative Godunov method for the advection system and solved following a coupled solution strategy. The resulting first-order accurate method is extended to second order of accuracy in space and time via the ADER approach together with an AENO reconstruction technique. Accuracy, robustness and well-balanced properties of the resulting scheme are assessed through a carefully selected suite of testcases. The scheme is exceedingly simple, accurate and robust as the sophisticated Godunov methods. A distinctive feature of the novel scheme is its flexibility in the choice of the sediment transport closure formula, which makes it particularly attractive for scientific and engineering applications.

Hybrid Multifluid Algorithms Based on the Path-Conservative Central-Upwind Scheme

Hybrid Multifluid Algorithms Based on the Path-Conservative Central-Upwind Scheme

Полные решатели Римана для гиперболической GPR модели механики сплошной среды

In this paper, complete Riemann solver of Osher-Solomon and the HLLEM Riemann solver for unified first order hyperbolic formulation of continuum mechanics, which describes both of fluid and solid dynamics, are presented. This is the first time that these types of Riemann solvers are applied to such a complex system of governing equations as the GPR model of continuum mechanics. The first order hyperbolic formulation of continuum mechanics recently proposed by Godunov S. K., Peshkov I. M. and Romenski E. I., further denoted as GPR model includes a hyperbolic formulation of heat conduction and an overdetermined system of PDE. Path-conservative schemes are essential in order to give a sense to the non-conservative terms in the weak solution framework since governing PDE system contains non-conservative products, too. New Riemann solvers are implemented and tested successfully, which means it certainly acts better than standard local Lax-Friedrichs-type or Rusanov-type Riemann solvers. Two simple computational examples are presented, but the obtained computational results clearly show that the complete Riemann solvers are less dissipative than the simple Rusanov method that was employed in previous work on the GPR model.

A path conservative finite volume method for a shear shallow water model

The shear shallow water model provides an approximation for shallow water flows by including the effect of vertical shear in the model. This model can be derived from the depth averaging process by including the second order velocity fluctuations which are neglected in the classical shallow water approximation. The resulting model has a non-conservative structure which resembles the 10-moment equations from gas dynamics. This structure facilitates the development of path conservative schemes and we construct HLL, 3-wave and 5-wave HLLC-type solvers. An explicit and semi-implicit MUSCL-Hancock type second order scheme is proposed for the time integration. Several test cases including roll waves show the performance of the proposed modeling and numerical strategy.

On a class of genuinely 2D incomplete Riemann solvers for hyperbolic systems

We propose a general class of genuinely two-dimensional (2D) incomplete Riemann solvers for hyperbolic systems of equations. In particular, extensions of the multidimensional HLL solver proposed by Balsara to 2D Polynomial Viscosity Matrix (PVM) finite volume methods are considered. The numerical flux is constructed by assembling, at each edge of the computational mesh, a one-dimensional PVM flux with two purely 2D PVM fluxes defined at corners, which take into account transversal features of the flow through the approximate solution of 2D Riemann problems. The proposed methods are applicable to general hyperbolic systems of conservation laws. A first version of the scheme was recently introduced in our other work, where applications to magnetohydrodynamics were considered, including an efficient technique for divergence cleaning of the magnetic field based on the nonconservative form of the ideal magnetohydrodynamics equations, which provides good results in combination with our 2D solvers. In this work, we focus on applications to shallow water systems, in which the source term due to the bottom topography introduces an additional difficulty. An elegant way to overcome this difficulty consists in reformulating the problem in nonconservative form. For this reason, we have extended our 2D schemes to the case of nonconservative hyperbolic systems, within the framework of path-conservative schemes introduced in the work of Parés. Again, the proposed schemes are applicable to general nonconservative hyperbolic systems. A number of challenging numerical experiments including genuinely 2D effects are presented to test the performances and advantages of the proposed schemes.

Path-conservative central-upwind schemes for nonconservative hyperbolic systems

We develop path-conservative central-upwind schemes for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. Such systems arise in a variety of applications and the most challenging part of their numerical discretization is a robust treatment of nonconservative product terms. Godunov-type central-upwind schemes were developed as an efficient, highly accurate and robust ``black-box’’ solver for hyperbolic systems of conservation and balance laws. They were successfully applied to a large number of hyperbolic systems including several nonconservative ones. To overcome the difficulties related to the presence of nonconservative product terms, several special techniques were proposed. However, none of these techniques was sufficiently robust and thus the applicability of the original central-upwind schemes was rather limited. In this paper, we rewrite the central-upwind schemes in the form of path-conservative schemes. This helps us (i) to show that the main drawback of the original central-upwind approach was the fact that the jump of the nonconservative product terms across cell interfaces has never been taken into account and (ii) to understand how the nonconservative products should be discretized so that their influence on the numerical solution is accurately taken into account. The resulting path-conservative central-upwind scheme is a new robust tool for both conservative and nonconservative hyperbolic systems. We apply the new scheme to the Saint-Venant system with discontinuous bottom topography and two-layer shallow water system. Our numerical results illustrate the good performance of the new path-conservative central-upwind scheme, its robustness and ability to achieve very high resolution.

HLLC-type and path-conservative schemes for a single-velocity six-equation two-phase flow model: A comparative study

The present article deals with the numerical integration of a six-equation single-velocity two-phase flow model with stiff mechanical relaxation. This model can be employed to approximate efficiently the well known single-velocity single-pressure five-equation model of Kapila et al. (2001). Work in the literature has shown the efficiency of the six-equation model in simulating complex two-phase flows involving cavitation and evaporation processes. The aim of this work is to present and discuss various numerical schemes for this two-phase model focusing on the integration of the nonconservative terms appearing in the phasic energy equations. In fact, previous work has suggested that the choice of the discretization method for the nonconservative terms often does not play a significant role. Two new methods are proposed: a path-conservative HLLC-type scheme that is based on the Dal Maso–LeFloch–Murat theory, and a generalized HLLC-type scheme that is based on a Suliciu’s Riemann solver. The latter scheme has the important property of preserving the positivity of the intermediate states of the conserved quantities. Moreover, we also approximate solutions of the six-equation model by applying two path-conservative schemes recently proposed in the literature, which have been derived from the Osher and HLLEM Riemann solvers. We show comparisons of the different numerical schemes for several test cases, including cavitation problems and shock tubes. An efficiency study for first and second order schemes is also presented. Numerical results show that different methods corresponding to different numerical treatments of the nonconservative terms give analogous results and they are all able to produce accurate approximations of solutions of the Kapila’s five-equation model, except, as expected, for shocks in two-phase mixtures with very high pressure ratios.

Reprint of: Well-balanced methods for the shallow water equations in spherical coordinates

The goal of this work is to obtain a family of explicit high order well-balanced methods for the shallow water equations in spherical coordinates. Application of shallow water models to large scale problems requires the use of spherical coordinates: this is the case, for instance, of the simulation of the propagation of a Tsunami wave through the ocean. Although the PDE system is similar to the shallow water equations in Cartesian coordinates, new source terms appear. As a consequence, the derivation of high order numerical methods that preserve water at rest solutions is not as straightforward as in that case. Finite Volume methods are considered based on a first order path-conservative scheme and high order reconstruction operators. Numerical methods based on these ingredients have been successfully applied previously to the nonlinear SWEs in Cartesian coordinates. Some numerical tests to check the well-balancing and high order properties of the scheme, as well as its ability to simulate planetary waves or tsunami waves over realistic bathymetries are presented.

Efficient analytical implementation of the DOT Riemann solver for the de Saint Venant–Exner morphodynamic model

Within the framework of the de Saint Venant equations coupled with the Exner equation for morphodynamic evolution, this work presents a new efficient implementation of the Dumbser-Osher-Toro (DOT) scheme for non-conservative problems. The DOT path-conservative scheme is a robust upwind method based on a complete Riemann solver, but it has the drawback of requiring expensive numerical computations. Indeed, to compute the non-linear time evolution in each time step, the DOT scheme requires numerical computation of the flux matrix eigenstructure (the totality of eigenvalues and eigenvectors) several times at each cell edge. In this work, an analytical and compact formulation of the eigenstructure for the de Saint Venant–Exner (dSVE) model is introduced and tested in terms of numerical efficiency and stability. Using the original DOT and PRICE-C (a very efficient FORCE-type method) as reference methods, we present a convergence analysis (error against CPU time) to study the performance of the DOT method with our new analytical implementation of eigenstructure calculations (A-DOT). In particular, the numerical performance of the three methods is tested in three test cases: a movable bed Riemann problem with analytical solution; a problem with smooth analytical solution; a test in which the water flow is characterised by subcritical and supercritical regions. For a given target error, the A-DOT method is always the most efficient choice. Finally, two experimental data sets and different transport formulae are considered to test the A-DOT model in more practical case studies.

Well-balanced methods for the shallow water equations in spherical coordinates

The goal of this work is to obtain a family of explicit high order well-balanced methods for the shallow water equations in spherical coordinates. Application of shallow water models to large scale problems requires the use of spherical coordinates: this is the case, for instance, of the simulation of the propagation of a Tsunami wave through the ocean. Although the PDE system is similar to the shallow water equations in Cartesian coordinates, new source terms appear. As a consequence, the derivation of high order numerical methods that preserve water at rest solutions is not as straightforward as in that case. Finite Volume methods are considered based on a first order path-conservative scheme and high order reconstruction operators. Numerical methods based on these ingredients have been successfully applied previously to the nonlinear SWEs in Cartesian coordinates. Some numerical tests to check the well-balancing and high order properties of the scheme, as well as its ability to simulate planetary waves or tsunami waves over realistic bathymetries are presented.

A well-balanced path conservative SPH scheme for nonconservative hyperbolic systems with applications to shallow water and multi-phase flows

Abstract The present paper is concerned with the development of a new path-conservative meshless Lagrangian particle method (SPH), for the solution of non-conservative systems of hyperbolic partial differential equations, with applications to the shallow water equations, compressible and incompressible multi-phase flows. For shallow water flows, the proposed method is well-balanced. The starting point of our work is the SPH formulation of Vila and Ben Moussa. The method is based on Arbitrary-Lagrangian-Eulerian (ALE) numerical flux functions (Riemann solvers) and the scheme is rewritten in flux-difference form, thus ensuring at least zeroth order consistency for constant solutions at the discrete level. Furthermore, a smoothed velocity field is used, in order to obtain a final particle velocity that is consistent with the chosen ALE interface velocity. Starting from the formulation previously described, we then use a path-conservative discretization of the non-conservative terms, following the pioneering work of Castro and Pares. The scheme uses a generalized Roe-matrix that is computed as the path integral of the non-conservative terms along a prescribed integration path, which in this paper is chosen to be a simple straight-line segment. For the shallow water systems under consideration, the segment path leads to well-balanced schemes that exactly preserve the water at rest solution for arbitrary bottom topography. To the knowledge of the authors, this is the first well-balanced SPH scheme based on approximate Riemann solvers and the framework of path-conservative schemes for non-conservative hyperbolic systems. Three different approximate Riemann solvers have been investigated and compared in this paper: the Rusanov scheme, the Osher-type DOT scheme and the HLLEM method. A detailed comparison concerning the accuracy and computational efficiency of each Riemann solver is provided. Several test problems (both in 1D and 2D) are presented in the final part of the paper for different systems of non-conservative hyperbolic equations: the Baer Nunziato model of compressible multi-phase flows, the one- and two-layer shallow water equations and the Pitman & Le two-phase debris flow model. The numerical results for each system of equations have been compared with available reference solutions to verify the accuracy of the proposed scheme. For the shallow water type models under consideration the method proposed in this paper has furthermore been shown to be well-balanced up to machine precision.

A new comment on the computation of non-conservative products using Roe-type path conservative schemes

We are interested in the numerical approximation of the discontinuous solutions of non-conservative hyperbolic systems. We more precisely consider a non-conservative formulation of the usual gas dynamics equations and show how to slightly modify the so-called Roe-type path-conservative schemes to properly capture the underlying shock discontinuities. Numerical evidences are proposed. The present note follows a first comment on the computation of non-conservative products in [1].

Momentum balance in the shallow water equations on bottom discontinuities

This work investigates the topical problem of balancing the shallow water equations over bottom steps of different heights. The current approaches in the literature are essentially based on mathematical analysis of the hyperbolic system of balance equations and take into account the relevant progresses in treating the non-conservative form of the governing system in the framework of path-conservative schemes. An important problem under debate is the correct position of the momentum balance closure when the bottom elevation is discontinuous. Cases of technical interest are systematically analysed, consisting of backward-facing steps and forward-facing steps, tackled supercritical and subcritical flows; critical (sonic) conditions are also analysed and discussed.The fundamental concept governing the problem and supported by the present computations is that the energy-conserving approach is the only approach that is consistent with the classical shallow water equations formulated with geometrical source terms and that the momentum balance is properly closed if a proper choice of a conventional depth on the bottom step is performed. The depth on the step is shown to be included between the depths just upstream and just downstream of the step. It is also shown that current choices (as given in the literature) of the depth on (or in front of) the step can lead to unphysical configurations, similar to some energy-increasing solutions.

Solving Two-Mode Shallow Water Equations Using Finite Volume Methods

AbstractIn this paper, we develop and study numerical methods for the two-mode shallow water equations recently proposed in [S. STECHMANN, A. MAJDA, and B. KHOUIDER, Theor. Comput. Fluid Dynamics, 22 (2008), pp. 407-432]. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches—two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme—and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method.

High order space–time adaptive ADER-WENO finite volume schemes for non-conservative hyperbolic systems

We present a class of high order finite volume schemes for the solution of non-conservative hyperbolic systems that combines the one-step ADER-WENO finite volume approach with space–time adaptive mesh refinement (AMR). The resulting algorithm, which is particularly well suited for the treatment of material interfaces in compressible multi-phase flows, is based on: (i) high order of accuracy in space obtained through WENO reconstruction, (ii) a high order one-step time discretization via a local space–time discontinuous Galerkin predictor method, and (iii) the use of a path conservative scheme for handling the non-conservative terms of the equations. The AMR property with time accurate local time stepping, which has been treated according to a cell-by-cell strategy, strongly relies on the high order one-step time discretization, which naturally allows a high order accurate and consistent computation of the jump terms at interfaces between elements using different time steps. The new scheme has been successfully validated on some test problems for the Baer–Nunziato model of compressible multiphase flows.

Application of Osher and PRICE-C schemes to solve compressible isothermal two-fluid models of two-phase flow

In this paper, two path-conservative schemes, namely Osher and PRICE-C schemes have been used to solve isothermal compressible two-phase flows using four-equation model. Path-Conservative Osher (PC-Osher) scheme is an upwind method using the full eigenstructure of the system but PRICE-C scheme is a central method, in which using the full eigenstructure of the system is not necessary. Different two-phase flow problems are solved using these schemes and their results are compared. The numerical efficiency of two schemes and their abilities in the simulation of near single phase flows are also examined. The extension of these schemes to the second order of accuracy is performed using the well-known TVD-MUSCL–Hancock (TMH) method. The results show that for the same level of accuracy, the PC-Osher is more efficient than the PRICE-C scheme in view of computational time. However, the PC-Osher scheme fails to predict near single phase flows compared to the PRICE-C scheme. The results also show that the second order extension of both schemes is less diffusive on the sonic waves while they show small amplitude oscillations on the volume fraction waves.

Slug type hydrodynamic instability analysis using a five equations hyperbolic two-pressure, two-fluid model

This paper presents a new and accurate method for modeling dynamic and transient behaviors of slug initiation and growth in horizontal ducts. In this analysis, a hyperbolic, two-pressure, five-equation, two-fluid model predicted the dynamic behavior of a complicated slug flow regime. The model was modified by including friction with the wall and at the interfaces between phases. In this paper, highly accurate shock capturing numerical methods were applied to slug modeling for the first time. The applied method is low cost, simple and does not exhibit the problems of previous numerical methods, such as the high computational time associated with the finite volume of two-fluid models. The new method does not use experimental correlations and assumptions to simplify the model like unit cell and slug tracking methods. In the present work, the numerical method of AUSMDV was used for a one-dimensional, two-pressure, five-equation, hyperbolic, two-fluid model. The calculation is conducted by applying second order time and space accuracy. The AUSMDV numerical method does not identify a complicated matrix solution; therefore, it renders the flow field solution using minimal computational resources. Discretization of non-conservative products that appear in the momentum equations was performed using a conservative linear path scheme (based on path conservative schemes by Pares), and accurate discretization of non-conservative terms based on AUSM fluxes have been conducted. Numerical solutions are presented for well-defined problems of two-phase, gas–liquid flows. The well-defined test cases that were considered for verification and validation are the Reiman Shock Tube and the Ransom Water Faucet. Numerical estimates of slug flows in horizontal ducts were compared with two sets of experimental results. Good agreement between modeled and experimental results, as well as a grid independency study, suggests that the presented model is capable of slug tracking and slug capturing. In addition, the numerical method that is used here can predict flows with sufficient accuracy. Finally, several facts related to the physics of slug behavior were outlined. These facts could be useful for scientists who wish to model the flow regime correctly, and they may be important for system designers who must predict slug behavior physically.

WENO Scheme with Subcell Resolution for Computing Nonconservative Euler Equations with Applications to One-Dimensional Compressible Two-Medium Flows

High order path-conservative schemes have been developed for solving nonconservative hyperbolic systems in (Pares, SIAM J. Numer. Anal. 44:300---321, 2006; Castro et al., Math. Comput. 75:1103---11...