Steady State Computations for Wave Propagation Problems

Abstract

Abstract. The behavior of difference approximations of hyperbolic partial differential equations as time t -* oo is studied. The rate of convergence to steady state is analyzed theoretically and expe ¡mentally for the advection equation and the linearized Euler equations. The choice of difference formulas and boundary conditions strongly influences the rate of convergence in practical steady state calculations. In particular it is shown that upwind difference methods and characteristic boundary conditions have very attractive convergence properties.1. Introduction. For computing steady state solutions to problems in fluid mechanics, the time-dependent formulation is often used. There are several mechanisms that drive the solution to a steady state. In this paper we shall concentrate on the dissipation effect due to the boundary conditions, and not to the effect of friction and viscosity. Therefore we shall study hyperbolic partial differential equations where the boundary effects are dominant. The results are also valid for more general classes of differential equations of essentially hyperbolic character, as for example the Navier-Stokes equations for high Reynolds numbers.The purpose of this paper is to analyze the convergence properties to steady state both for the continuous problem and the corresponding discrete approximation. The basis for this study is the behavior of the spectrum of the differential and the difference operators, respectively. Two model problems are considered, the scalar advection problem and the isentropic Euler equation problem. The latter problem is studied in a two-dimensional geometry corresponding to channel flow. It is shown that the choice of boundary conditions radically affects the convergence rate to steady state as time increases. The asymptotic rate may actually change from exponential to algebraic, and for certain sets of boundary conditions there may be no convergence at all.The convergence rate for time-marching procedures has been discussed by others. For example, in [4] Giles investigated the eigenmodes of the solution of the one-dimensional Euler equations under various boundary conditions. Eriksson and Rizzi in [2] studied the convergence rate for a time-marching centered finite-volume

Variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations

“…It follows from (48), (49) and the correlation coefficients in (50) and (51) that the contribution to the decay of the variance depends on the correlation between the characteristic variables at the boundaries. A negative correlation lowers the variance and a positive correlation increases the variance.…”

“…Figure 3 and 4 shows the variance decay with time comparing characteristic and non-characteristic boundary conditions when using a normally (ξ ∼ N (0, 1)) and uniformly distributed (ξ ∼ U(− √ 3, √ 3)) ξ respectively. Note the similarity between studying variance decay in the case of zero variance on the boundary and analyzing convergence to steady state, see [49,50,51].…”

Summation-by-parts formulations for flow problems

“…In this section, normal mode analysis [17] is used to estimate the convergence rate to steady state for different sets of open boundary conditions for the INS equations. This procedure has been used in several studies, see for example [28,7], and the results in this section are found in Article III. A similar procedure is also used in Article I for the compressible Euler equations and Article II for the spatial operator of the INS equations.…”

Practical inlet boundary conditions for internal flow calculations

“…The solution to the whole system (23) becomes φ = (0, 0, 0, 0) and ψ = σ 3 ψ 3 + σ 4 ψ 4 , where ψ 3 = (− √ γ − 1, 0, 0, 1) and ψ 4 = (0, iωū, s, 0) . Finally, the homogeneous solution to (20) is Ŵh = ΨK(x) σ, where Ψ = [ψ 1 , ψ 2 , ψ 3 , ψ 4 ], K(x) = diag(e κ 1 , e κ 2 , e κ 3 , e κ 3 ) and σ = (σ 1 , σ 2 , σ 3 , σ 4 ) contain the solution vectors, the exponential functions and the parameters that will be determined from the boundary conditions, respectively.…”