Richard Sanders

High resolution staggered mesh approach for nonlinear hyperbolic systems of conservation laws

Abstract A numerical technique is presented to approximate weak solutions of hyperbolic systems of conservation laws in one and two space dimensions. When strong shocks are present in the exact solution other techniques rely on approximate Riemann problem solvers to cope with this difficulty. The high-order approach here does not use approximate Riemann solvers as a building block and is therefore considerably easier to implement compared to the usual methods. Moreover, the approach here yields sharp nonoscillatory shocks, sharp corners at the base of rarefaction waves and has high order accuracy in regions where the solution is smooth. For the scalar one-dimensional problem the theoretical results confirm the reliability of this approach.

SELF-SIMILAR SOLUTIONS FOR THE TRIPLE POINT PARADOX IN GASDYNAMICS ∗

We present numerical solutions of a two-dimensional Riemann problem for the com- pressible Euler equations that describes the Mach reflection of weak shock waves. High resolution finite volume schemes are used to solve the equations formulated in self-similar variables. We use extreme local grid refinement to resolve the solution in the neighborhood of an apparent but math- ematically inadmissible shock triple point. The solutions contain a complex structure: instead of three shocks meeting in a single standard triple point, there is a sequence of triple points and tiny supersonic patches behind the leading triple point, formed by the reflection of weak shocks and ex- pansion waves between the sonic line and the Mach shock. An expansion fan originates at each triple point, resolving the von Neumann triple point paradox.

THE TRIPLE POINT PARADOX FOR THE NONLINEAR WAVE SYSTEM

We present numerical solutions of a two-dimensional Riemann problem for the non- linear wave system which is used to describe the Mach reflection of weak shock waves. Robust low order as well as high resolution finite volume schemes are employed to solve this equation formulated in self-similar variables. These, together with extreme local grid refinement, are used to resolve the solution in the neighborhood of an apparent but mathematically inadmissible shock triple point. Rather than observing three shocks meeting in a single standard triple point, we are able to detect a primary triple point containing an additional wave, a centered expansion fan, together with a se- quence of secondary triple points and tiny supersonic patches embedded within the subsonic region directly behind the Mach stem. An expansion fan originates at each triple point. It is our opinion that the structure observed here resolves the von Neumann triple point paradox for the nonlinear wave system. These solutions closely resemble the solutions obtained in (A. M. Tesdall and J. K. Hunter, SIAM J. Appl. Math., 63 (2002), pp. 42-61) for the unsteady transonic small disturbance equation.

The Neumann problem for nonlinear second order singular perturbation problems

Singularly perturbed second order elliptic partial differential equations with Neumann boundary conditions arise in many areas of application. These problems rarely have smooth limit solutions. In this paper, we characterize the limit solution for a wide class of such problems. We also give an abstract rate of convergence theorem and apply the abstract theorem to certain finite difference approximations.

The Numerical Study of Singular Shocks Regularized by Small Viscosity

A singular shock is a measure valued solution found by considering viscosity limit solutions to certain hyperbolic systems. In this work, some fundamental properties concerning such solutions are derived and then illustrated by extensive numerical study.

Further Results on Guderley Mach Reflection and the Triple Point Paradox

Recent numerical solutions and shock tube experiments have shown the existence of a complex reflection pattern, known as Guderley Mach reflection, which provides a resolution of the von Neumann paradox of weak shock reflection. In this pattern, there is a sequence of tiny supersonic patches, reflected shocks and expansion waves behind the triple point, with a discontinuous transition from supersonic to subsonic flow across a shock at the rear of each supersonic patch. In some experiments, however, and in some numerical computations, a distinctly different structure which has been termed Guderley reflection has been found. In this structure, there appears to be a single expansion fan at the triple point, a single supersonic patch, and a smooth transition from supersonic to subsonic flow at the rear of the patch. In this work, we present numerical solutions of the compressible Euler equations written in self-similar coordinates at a set of parameter values that were used in previous computations which found the simple single patch structure described above. Our solutions are more finely resolved than these previous solutions, and they show that Guderley Mach reflection occurs at this set of parameter values. These solutions lead one to conjecture that the two patterns are not distinct: rather, Guderley reflection is actually underresolved Guderley Mach reflection.

On the rate of convergence of viscosity solutions for boundary value problems

A class of singularly perturbed boundary value problems is considered for viscosity tending to zero. From compactness arguments it is known that the solutions converge to a limit function characterized by an entropy inequality. We formulate an approximate entropy inequality (AEI) and use it to obtain the order of convergence. The AEI is also used to obtain the order of convergence for monotone difference schemes.

Second order nonlinear singular perturbation problems with boundary conditions of mixed type

We study nonlinear turning point problems that admit boundary and/or interior layers at positions that are not determined a priori. Our study differs from previous investigations in that for positive “viscosity” first order derivative terms are allowed in the boundary operator. Under certain conditions, shown in a sense to be sharp, we characterize the viscous limit of such problems and prove that they are identical to those limit solutions obtained from the pure Dirichlet problem.

The von Neumann Triple Point Paradox

We describe the problem of weak shock reflection off a wedge and discuss the triple point paradox that arises. When the shock is sufficiently weak and the wedge is thin, Mach reflection appears to be observed but is impossible according to what von Neumann originally showed in 1943. We summarize some recent numerical results for weak shock reflection problems for the unsteady transonic small disturbance equations, the nonlinear wave system, and the Euler equations. Rather than finding a standard but mathematically inadmissible Mach reflection with a shock triple point, the solutions contain a complex structure: there is a sequence of triple points and supersonic patches in a tiny region behind the leading triple point, with an expansion fan originating at each triple point. The sequence of patches may be infinite, and we refer to this structure as Guderley Mach reflection. The presence of the expansion fans at the triple points resolves the paradox. We describe some recent experimental evidence which is consistent with these numerical findings.

Carbuncle phenomenon: On upwind schemes in multidimensions

Abstract The purpose of this note is to provide some insight on and a cure to the well-known ‘carbuncle phenomenon’. The usual dimension by dimension extension of upwind schemes designed for the one-dimensional equations of gas dynamics to the multidimensional equations yields poorly resolved stationary (or slowly moving) shocks when applied to high Mach number grid aligned flows on structured grids. We suggest that this phenomenon is due to inadequate crossflow dissipation offered by the usual approach. Here, we propose a parameter-free and easy to implement multidimensional, upwind dissipation that provides sufficient crossflow dissipation to alleviate the carbuncle pathology. The proposed approach also retains the desirable properties offered by one-dimensional upwinding. Two numerical examples are considered for the purpose of comparison.