Saul Abarbanel

Optimal time splitting for two- and three-dimensional navier-stokes equations with mixed derivatives

A new explicit, time splitting algorithm has been developed for finite difference modelling of the full two and three-dimensional time-dependent, compressible, viscous Navier-Stokes equations of fluid mechanics. The scheme is optimal in the sense that the split operators achieve their maximum allowable time step, i.e., the corresponding Courant number. The algorithm allows a conservation-form formulation. Stability is proven analytically and verified numerically. In proving stability it was shown that all nine matrix coefficients of the Navier-Stokes equations are simultaneously symmetrizable by a similarity transformation. Two such transformations and their resulting symmetric matrix coefficients are presented explicitly.

The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error

The conventional method of imposing time dependent boundary conditions for Runge-Kutta (RK) time advancement reduces the formal accuracy of the space-time method to first order locally, and second order globally, independently of the spatial operator. This counter intuitive result is analyzed in this paper. Two methods of eliminating this problem are proposed for the linear constant coefficient case: 1) impose the exact boundary condition only at the end of the complete RK cycle, 2) impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives. The first method, while retaining the RK accuracy in all cases , results in a scheme with much reduced CFL condition, rendering the RK scheme less attractive. The second method retains the same allowable time step as the periodic problem. However it is a general remedy only for the linear case. For non-linear hyperbolic equations the second method is effective only for for RK schemes of third order accuracy or less. Numerical studies are presented to verify the efficacy of each approach.

On the construction and analysis of absorbing layers in CEM

Abstract A recently introduced system of partial differential equations, based on physical considerations, which describes the behavior of electro-magnetic waves in artificial absorbing layers, is analyzed. Analytic solutions are found, for the cases of semi-infinite layers and finite depth layers, both for primitive and characteristic boundary conditions. A different set of equations that seem to offer some advantages is proposed in this paper. The properties of its solutions for the same geometries and boundary condition are also discussed.

Long Time Behavior of the Perfectly Matched Layer Equations in Computational Electromagnetics

We investigate the long time behavior of two unsplit PML methods for the absorption of electromagnetic waves. Computations indicate that both methods suffer from a temporal instability after the fields reach a quiescent state. The analysis reveals that the source of the instability is the undifferentiated terms of the PML equations and that it is associated with a degeneracy of the quiescent systems of equations. This highlights why the instability occurs in special cases only and suggests a remedy to stabilize the PML by removing the degeneracy. Computational results confirm the stability of the modified equations and is used to address the efficacy of the modified schemes for absorbing waves.

Stable and accurate boundary treatments for compact, high-order finite-difference schemes

Abstract The stability characteristics of various compact fourth- and sixth-order spatial operators sre used to assess the theory of Gustafsson, Kreiss and Sundstrom (G-K-S) for the semidiscrete initial boundary value problem (IBVP). In all cases, favorable comparisons are obtained between G-K-S theory, eigenvalue determination, and numerical simulation. The conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable (bounded, left half-plane (LHP) eigenvalues). It is shown that many of the higher-order schemes that are G-S-K stable are not asymptotically stable. A series of compact fourth- and sixth-order schemes is developed, all of which are asymptotically and G-K-S stable for the scalar case. A systematic technique is then presented for constructing stable and accurate boundary closures of various orders. The technique uses the semidescrete summation-by-parts energy norm to guarantee asymptotic and G-K-S stability of the resulting boundary closure. Various fourth-order explicit and implicit discretizations are presented, all of which satisfy the summation-by-parts energy norm.

Compact high-order schemes for the Euler equations

An implicit approximate factorization (AF) algorithm is constructed, which has the following characteristics.• In two dimensions: The scheme is unconditionally stable, has a 3×3 stencil and at steady state has a fourth-order spatial accuracy. The temporal evolution is time accurate either to first or second order through choice of parameter.• In three dimensions: The scheme has almost the same properties as in two dimensions except that it is now only conditionally stable, with the stability condition (the CFL number) being dependent on the “cell aspect ratios,”Δy/Δx andΔz/Δx. The stencil is still compact and fourth-order accuracy at steady state is maintained.Numerical experiments on a two-dimensional shock-reflection problem show the expected improvement over lower-order schemes, not only in accuracy (measured by theL2 error) but also in the dispersion. It is also shown how the same technique is immediately extendable to Runge-Kutta type schemes, resulting in improved stability in addition to the enhanced accuracy.

Strict Stability of High-Order Compact Implicit Finite-Difference Schemes

Temporal, or “strict,” stability of approximation to PDEs is much more difficult to achieve than the “classical” Lax stability. In this paper, we present a class of finite-difference schemes for hyperbolic initial boundary value problems in one and two space dimensions that possess the property of strict stability. The approximations are constructed so that all eigenvalues of corresponding differentiation matrix have a nonpositive real part. Boundary conditions are imposed by using penalty-like terms. Fourth- and sixth-order compact implicit finite-difference schemes are constructed and analyzed. Computational efficacy of the approach is corroborated by a series of numerical tests in 1-D and 2-D scalar problems.

On the Removal of Boundary Errors Caused by Runge--Kutta Integration of Nonlinear Partial Differential Equations

The temporal integration of hyperbolic partial differential equations (PDEs) has been shown to lead sometimes to the deterioration of accuracy of the solution because of boundary conditions. A procedure for removal of this error in the linear case has been established previously. In this paper we consider hyperbolic PDEs (linear and nonlinear) whose boundary treatment is accomplished via the simultaneous approximation term (SAT) procedure. A methodology is presented for recovery of the full order of accuracy and has been applied to the case of a fourth-order explicit finite-difference scheme.

External flow computations using global boundary conditions

We numerically integrate the compressible Navier-Stokes equations by means of a finite volume technique on the domain exterior to an airfoil. The curvilinear grid we use for discretization of the Navier-Stokes equations is obviously finite, it covers only a certain bounded region around the airfoil, consequently, we need to set some artificial boundary conditions at the external boundary of this region. The artificial boundary conditions we use here are non-local in space. They are constructed specifically for the case of steady-state solution. In constructing the artificial boundary conditions, we linearize the Navier-Stokes equations around the far-field solution and apply the difference potentials method. The resulting global conditions are implemented together with a pseudotime multigrid iteration procedure for achieving the steady state. The main goal of this paper is to describe the numerical procedure itself, therefore, we primarily emphasize the computation of artificial boundary conditions and the combined usage of these artificial boundary conditions and the original algorithm for integrating the Navier-Stokes equations. The underlying theory that justifies the proposed numerical techniques will accordingly be addressed more briefly.

Splitting methods for low Mach number Euler and Navier-Stokes equations

Abstract In this paper, we examined some splitting techniques for low Mach number Euler flows. We point out shortcomings of some of the proposed methods and suggest an explanation for their inadequacy. We then present a symmetric splitting for both the Euler and Navier-Stokes equations which removes and stiffness of these equations when the Mach number is small. The splitting is shown to be stable.