Matania Ben-Artzi

A second-order Godunov-type scheme for compressible fluid dynamics

Abstract A second-order accurate scheme for the integration in time of the conservation laws of compressible fluid dynamics is presented. Two related versions are proposed, one Lagrangian and the second direct Eulerian. They both share the common ingredient which is a full analytic solution for the time derivatives of flow quantities at a jump discontinuity, assuming initial nonvanishing slopes on both sides. While this solution is an extension of the solution to the classical Riemann problem, the resulting schemes are second-order extensions of Godunovs methods. In both cases, they are very simple to implement in computer codes. Several numerical examples are shown, where the only additional mechanism is a simple monotonicity algorithm.

Unitary equivalence and scattering theory for Stark‐like Hamiltonians

Let H0=−Δ+V0( x1), H=H0+V( x) be self‐adjoint in L2(Rn). V0 depends only on one coordinate and tends monotonically to ∓∞ as x1→±∞. V is a real H0‐compact potential, short range with respect to V0. In particular, the cases V0( x1)=−(sgn x1)‖x1‖α, 0<α≤2 and ‖V( x)‖≤C‖x‖−1 are included (α=1 being the Stark effect). It is shown that (a) H is spectrally absolutely continuous over the entire real axis apart from a possible discrete sequence of eigenvalues of finite multiplicity and rapidly decaying eigenfunctions (H0 has no eigenvalues) and (b) the wave operators W±(H,H0) exist and are complete.

A limiting absorption principle for Schrödinger operators with spherically symmetric exploding potentials

LetH=−Δ+V(r) be a Schrödinger operator with a spherically symmetric exploding potential, namely,V(r)=VS(r)+VL(r), whereVS(r) is short-range and the exploding partVL(r) satisfies the following assumptions: (a) Λ=lim supr→∞VL(r)<∞ (but Λ=−∞ is possible). Denote Λ+= max(Λ,0). (b)VL(r)∈C2k (r0, ∞) and, with someδ>0 such that 2kδ>1: (d/dr)jVL(r) · (Λ+−VL(r))−1=O(rjδ) asr → ∞,j=1, ..., 2k. (c) ∫r0∞dr|VL(r|1/2dr|VL(r)|1/2=∞. (d) (d/dr)VL(r)≦0. Under these assumptions a limiting absorption principle forR(z)=(H−z)−1 is established. More specifically, ifK ⊆C+={zImz≧0} is compact andK ∩ (−∞, Λ]=Ø thenR (z) can be extended as a continuous map ofK intoB (Y, Y*) (with the uniform operator topology), whereY ⊆L2(Rn) is a weighted-L2 space. To ensure uniqueness of solutions of (H−z)u=f, z ∈K, a suitable radiation condition is introduced.

Remarks on Schrödinger operators with an electric field and deterministic potentials

Abstract Let H 0 = − d 2 dx 2 + F · x, H = H 0 + V be Schrodinger operators on the line, where F ≠ 0 is a real constant and V a real potential. The case where V is unbounded and non-smooth is studied. It is shown that for a large class of potentials H is purely absolutely continuous and in fact unitarily equivalent to H0.

On the Uniqueness ofL2-solutions in half-space of certain differential equations

Square integrable solutions to the equation{−∂2/∂y2 + P(Dx)+b(y)−λ}u(x, y) = f(x, y) are considered in the half-spacey>0, x ∈ℝn, whereP(Dx) is a constant coefficient operator. Under suitable conditions on limy→0u(x, y), b(y), f(x, y) and λ, it is shown that suppu = suppf. This generalizes a result due to Walter Littman.

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].

A Limiting Absorption Principle for a Sum of Tensor Products

A limiting absorption principle is obtained for a sum of tensor products of the form under various hypotheses on the self-adjoint operators H1 and H2.