Joseph Falcovitz

Application of the GRP scheme to open channel flow equations

The GRP (generalized Riemann problem) scheme, originally conceived for gasdynamics, is reformulated for the numerical integration of the shallow water equations in channels of rectangular cross-section, variable width and bed profile, including a friction model for the fluid-channel shear stress. This scheme is a second-order analytic extension of the first-order Godunov-scheme, based on time-derivatives of flow variables at cell-interfaces resulting from piecewise-linear data reconstruction in cells. The second-order time-integration is based on solutions to generalized Riemann problems at cell-interfaces, thus accounting for the full governing equations, including source terms. The source term due to variable bed elevation is treated in a well-balanced way so that quiescent flow is exactly replicated; this is done by adopting the Surface Gradient Method (SGM). Several problems of steady or unsteady open channel flow are considered, including the terms corresponding to variable channel width and bed elevation, as well as to shear stress at the fluid-channel interface (using the Manning friction model). In all these examples remarkable agreement is obtained between the numerical integration and the exact or accurate solutions.

Applying the SMG Scheme to Reactive Flow

The staggered mesh Godunov-SMG scheme for Lagrangian and Arbitrary Lagrangian Eulerian (ALE) hydrodynamics has several potential advantages for applications involving reactive flow simulations arising in the initiation and propagation of detonation. This includes the capabilities to capture discontinuities present in an expanding flow and its inherent hourglass damping property. In the current work, we add to the stagered mesh Godunov (SMG) scheme an appropriate reaction rate law and an equation of state for mixtures of reactants and reaction products, and we test the performance of the scheme to simulate the detonation initiation and propagation over an initially deformed mesh.

Staggered Mesh Godunov (SMG) Schemes for Lagrangian Hydrodynamics

The difficulties inherent in converting the zone‐centered Godunov method into a 3D Lagrangian/ALE scheme have led us to propose an SMG scheme. The SMG/Q version presented here solves internal energy and momentum equations by using only zone‐centered “collision” Riemann problems. It is formulated in a dual Godunov/Classical‐Lagrange way. A limited‐slope approximation of zone‐centered velocity gradients produces a second‐order extension of this method. Basic test cases, both 1D and 3D, demonstrate the SMG/Q features.

Remarks on High-Resolution Split Schemes Computation

The high-resolution generalized Riemann problem (GRP) conservation laws scheme for compressible flows combined with Strang-type operator splitting is applied to computing an initial value problem having a discontinuous initial data. Imperfect representation of the initial data on the Cartesian grid, where the smooth curve of discontinuity is approximated by a jagged line, gives rise to spurious waves when using high-resolution integration with operator splitting. The nature of these waves is clarified by comparison to a one-dimensional model. We demonstrate that it is not the operator splitting that gives rise to these waves, but rather the better quality of the hyperbolic (one-dimensional) solver, which is not degraded by the operator splitting. It is expected that this property of retaining sharp features of initial data will also be produced by other second-order conservation laws schemes.

The Generalized Riemann Problem (GRP) for Compressible Fluid Dynamics

This chapter is concerned with the main topic of the monograph, namely, the solution of the GRP for quasi-1-D, inviscid, compressible, nonisentropic, timedependent flow. In Section 5.1 we formulate the problem and study its solution in the Lagrangian and Eulerian frames. In particular, we state and prove the main ingredient in the GRP method, Theorem 5.7. A weaker form of this theorem leads to the “acoustic approximation” (Proposition 5.9). Summary 5.24 gives a step-by-step description of the GRP analysis. In Section 5.2 we present the GRP methodology for the construction of second-order, high-resolution finite-difference (or finite-volume) schemes. Starting out from the (first-order) Godunov scheme, we present the basic (E1) GRP scheme. It is based on the acoustic approximation and constitutes the simplest second-order extension of Godunov’s scheme. This is followed by a presentation of the full array of GRP schemes (as well as MUSCL). Generally speaking, the presentation in this chapter follows closely the GRP papers [7] and [10].