Path-conservative Method Research Articles

A one-dimensional augmented Shallow Water Equations system for channels of arbitrary cross-section

This work provides a new formulation of the one-dimensional augmented Shallow Water Equations system for open channels and rivers with arbitrarily shaped cross sections, suitable for numerical integration when discontinuous geometry is encountered. The additional variable considered can be the bottom elevation, a reference width, a shape coefficient, or a vector containing these or other geometric parameters. The appropriate numerical method, which is well suited to coupling with the mathematical one, is a path-conservative method, capable of reconstructing the behaviour of physical and geometrical variables at the cell boundaries, where the discrete solution of hyperbolic systems of equations is discontinuous. A nonlinear path suitable for the shallow water context is adopted. The resulting model is shown to be well-balanced and accurate to the second order and is further validated against analytical solutions related to channels with power-law cross-sections, specifically for dam break patterns over a variable-width channel and the run-up dynamics of long water waves over sloping bays.

In-cell discontinuous reconstruction path-conservative methods for non conservative hyperbolic systems - Second-order extension

We are interested in the numerical approximation of discontinuous solutions in non conservative hyperbolic systems. An extension to second-order of a new strategy based on in-cell discontinuous reconstructions to deal with this challenging topic is presented. This extension is based on the combination of the first-order in-cell reconstruction with the standard MUSCL-Hancock reconstruction. The first-order strategy allowed in particular to capture exactly the isolated shocks and this new second-order extension keep this property. Moreover, the well-balanced property of the method is also studied. Several numerical tests are proposed to validate the methods for the Coupled-Burgers system, Gas dynamics equations in Lagrangian coordinates and the modified shallow water system.

FORCE schemes on moving unstructured meshes for hyperbolic systems

The aim of this paper is to propose a new simple and robust numerical flux of the centered type in the context of Arbitrary-Lagrangian–Eulerian (ALE) finite volume schemes. The work relies on the FORCE flux of Toro and Billet and is concerned with the solution of general hyperbolic systems of nonlinear equations involving both conservative and non-conservative terms as well as sources which might become stiff. The proposed approach is formulated in a general way using a path-conservative method and the Roe-type system matrix is computed numerically in order to provide a numerical flux function that can be applied to any given hyperbolic system. Furthermore, one great advantage of the FORCE flux is that no information about the eigenstructure of the system is needed, not even eigenvalues, but only information regarding the geometry of the control volumes are required, which are automatically available in the moving mesh framework. Our method is of the finite volume type, high order accurate in space, thanks to a WENO reconstruction operator, and even in time, due to a fully-discrete ADER one-step discretization. The algorithm applies to moving multidimensional unstructured meshes composed by triangles and tetrahedra. Both accuracy and robustness of the scheme are assessed on a series of test problems for the Euler equations of compressible gas dynamics, for the magnetohydrodynamics equations as well as for the Baer–Nunziato model of compressible multi-phase flows.

A Second-Order Path-Conservative Method for the Compressible Non-Conservative Two-Phase Flow

A Second-Order Path-Conservative Method for the Compressible Non-Conservative Two-Phase Flow

A flux splitting method for the Baer–Nunziato equations of compressible two-phase flow

Here we extend the Toro–Vázquez flux vector splitting approach (TV), originally proposed for the ideal 1D Euler equations in [1], to the Baer–Nunziato equations of compressible two-phase flow. Following the TV approach we identify corresponding advection and pressure operators. We perform a rigorous analysis of the associated non-conservative pressure system and derive its complete characteristic structure. The choice of the advection numerical flux is obvious. For the pressure system, several schemes are presented. The complete schemes are then implemented in the setting of finite volume and path-conservative methods and are systematically assessed in terms of accuracy and efficiency, through a carefully selected suite of test problems. The presented schemes constitute a building block for the construction of high-order numerical methods for solving the Baer–Nunziato equations. Here, as an illustrative example of such possibility, we present the construction of a second-order scheme.

A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

In this work, we present a class of fast first order finite volume solvers, called PVM (polynomial viscosity matrix), for balance laws or, more generally, for nonconservative hyperbolic systems. They are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe matrix. These methods have the advantage that they only need some information about the eigenvalues of the system to be defined, and no spectral decomposition of a Roe matrix is needed. As a consequence, they are faster than the Roe method. These methods can be seen as a generalization of the schemes introduced by Degond et al. in [C. R. Acad. Sci. Paris Sér. l Math., 328 (1999), pp. 479--483] for balance laws and nonconservative systems. The first order path conservative methods to be designed here are intended to be used as the basis for higher order methods for multidimensional problems. In this work, some well known solvers, such as Rusanov, Lax--Friedrichs, FORCE (see [E. F. Toro and S. J. Billett, IMA J. Numer. Anal., 20 (2000), pp. 47--79; M. J. Castro, A. Pardo, C. Parés, and E. F. Toro, Math. Comp., 79 (2010), pp. 1427--1472]), GFORCE (see [E. F. Toro and V. A. Titarev, J. Comput. Phys., 216 (2006), pp. 403--429]), and HLL (see [A. Harten, P. D. Lax, and B. van Leer, SIAM Rev., 25 (1983), pp. 35--61]), are redefined under this form, and then some new solvers are proposed. Finally, some numerical tests are presented, and the performances of the numerical schemes are compared with each others and with Roe scheme for the 1D and 2D two-fluid flow model of Pitman and Le.

A path-conservative method for a five-equation model of two-phase flow with an HLLC-type Riemann solver

Compressible multi-phase flows are found in a variety of scientific and engineering problems. The development of accurate and efficient numerical algorithms for multi-phase flow simulations remains one of the challenging issues in computational fluid dynamics. A main difficulty of numerical methods for multi-phase flows is that the model equations cannot always be written in conservative form, though they may be hyperbolic and derived from physical conservation principles. In this work, assuming a hyperbolic model, a path-conservative method is developed to deal with the non-conservative character of the equations. The method is applied to solve the five-equation model of Saurel and Abgrall for two-phase flow. As another contribution of the work, a simplified HLLC-type approximate Riemann solver is proposed to compute the Godunov state to be incorporated into the Godunov-type path-conservative method. A second order, semi-discrete version of the method is then constructed via a MUSCL reconstruction with Runge–Kutta time stepping. Moreover, the method is then extended to the two-dimensional case by directional splitting. The method is systematically assessed via a series of test problems with exact solutions, finding satisfactory results.

A Well-Balanced Path-Integral f-Wave Method for Hyperbolic Problems with Source Terms

Systems of hyperbolic partial differential equations with source terms (balance laws) arise in many applications where it is important to compute accurate time-dependent solutions modeling small perturbations of equilibrium solutions in which the source terms balance the hyperbolic part. The f-wave version of the wave-propagation algorithm is one approach, but requires the use of a particular averaged value of the source terms at each cell interface in order to be "well balanced" and exactly maintain steady states. A general approach to choosing this average is developed using the theory of path conservative methods. A scalar advection equation with a decay or growth term is introduced as a model problem for numerical experiments.

FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems

In this paper we propose a new high order accurate centered path-conservative method on unstructured triangular and tetrahedral meshes for the solution of multi-dimensional non-conservative hyperbolic systems, as they typically arise in the context of compressible multi-phase flows. Our path-conservative centered scheme is an extension of the centered method recently proposed in [40] for conservation laws, to which it reduces if the system matrix is the Jacobian of a flux function. The main advantage in the proposed centered approach compared to upwind methods is that no information about the eigenstructure of the system or Roe averages are needed. The final fully discrete high order accurate formulation in space and time is obtained using the general framework of P N P M schemes proposed in [16], which unifies in one single general family of schemes classical finite volume and discontinuous Galerkin methods. These P N P M methods can also be called reconstructed discontinuous Galerkin schemes, due to the use of the P N P M least-squares reconstruction operator. We show applications of our high order accurate unstructured centered method to the two- and three-dimensional Baer–Nunziato equations of compressible multiphase flows as introduced in [4].

ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows

We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step P N P M schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209–53]. The two key ingredients of our high order approach are: first, the high order accurate P N P M reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, Käser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204–43] to ensure monotonicity at discontinuities, and second, a local space–time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971–4001]. These two key ingredients are combined with the recently developed path-conservative methods of Parés [Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21] and Castro et al. [Castro MJ, Gallardo JM, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006;75:1103–34] to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le [Pitman EB, Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005;363:1573–601].

WELL-BALANCED NUMERICAL SCHEMES BASED ON A GENERALIZED HYDROSTATIC RECONSTRUCTION TECHNIQUE

The goal of this paper is to generalize the hydrostatic reconstruction technique introduced in Ref. 2 for the shallow water system to more general hyperbolic systems with source term. The key idea is to interpret the numerical scheme obtained with this technique as a path-conservative method, as defined in Ref. 35. This generalization allows us, on the one hand, to construct well-balanced numerical schemes for new problems, as the two-layer shallow water system. On the other hand, we construct numerical schemes for the shallow water system with better well-balanced properties. In particular we obtain a Roe method which solves exactly every stationary solution, and not only those corresponding to water at rest.