Technical approach

For simplicity, the analysis is performed for the one-dimensional wave equa­tion given by

(1)

Applying the dual-time stepping technique, a fictitious time, т, is introduced, and equation (1) is rewritten in the form

or

The derivative with the respect to the real time, t, is discretized as

/dun+1 aiun+1 + a,2Un + азип 1 + a^vl1 2 + … + am+2u" m

Ы = At ’ ()

where At is the physical time step, and the superscript n denotes the solution at physical time t = nAt. Equation (3) is marched in fictitious time, т, to reach a pseudo-steady state, which advances the solution forward in time from t = nAt to t = (n + 1)At. In order to speed up the convergence of the residual R*(u) to zero, acceleration techniques like local time stepping and multigridding are usually incorporated.

It is a common practice [Melson et al., 1993] to use a three-point backward formula to discretize the time derivative term, du/dt, i. e.

rdun+l 3u"+1 — 4u" + u"-1

Ы = 2ДЇ ’ ()

which is second order accurate in time and unconditionally stable. It can be shown that discretizations higher than second order accurate are only condi­tionally stable which makes them not practical. To improve the quality of the discretization (5) an optimization technique for constructing low-dissipation and low-dispersion schemes [Tam and Webb, 1993,Hu et al., 1994] is applied. Consider a Fourier representation of the solution, u, given by

u(r, t,x) = ^ ui (t )exp[i(u t — hi x)], (6)

і

where і = л/^І. Substituting (4) and (6) into (1) yields an approximation of the dispersion relation for the wave equation (1). This is given by

^(ai+a2 exp(—iu)At)+ct3 exp(—2iujAt)+a4 exp(—Зга; Af)+…) — ick = 0,

t (7)

or

u* — ck = 0, (8)

where u* = —i[a1 + a2 exp(—iuAt) + a3 exp(—2iuAt) + a4 exp(—3iuAt) + …]/At is a numerical approximation of the exact frequency, u. Equation (6) can be presented in the form

u(T, t,x) = ^ ul (T)exp(iui t)exp(—Re[iu*x/c]) exp(—ilm[iu* x/c]).

(9)

From equation (9) the dissipation (amplitude) and dispersion (phase) error are estimated by the equations

tampl = Re[iU*At — iuAt], (10)

{phase = 1m[iu*At — iuAt]. (11)

Optimized schemes can be constructed by introducing more points to the dis­cretization stencil (4) than the minimum number of points required to maintain the order of accuracy. These extra points are used to minimize the dissipation and dispersion errors given by (10) and (11).

In this paper, a four point second order optimized scheme is constructed. The coefficients, ai, i = 1,4, for this scheme are determined from the ac­curacy condition given by

ai = 1.5 — a4

a2 = — 2 + 3a4 (12)

a3 = 0.5 — 3a4

that is obtained by expanding the solution, u, in equation (4) into a Taylor series and keeping all the terms up to the second order. The free parameter, a 4, is found subject to constraints on the amplitude and phase errors,

—0.001 < eampi < 0.002, шДt < 0.7

tphase < 0.005, uAt < 0.3 ()

where the limits are set to keep the dissipation and dispersion errors smaller than those of the non-optimized second order scheme. The resulting coeffi­cients are given in Table 1. Figure 1 shows the stability plots and Fig. 2 shows the dissipation and dispersion errors. Note, that if Re(iw*Д^ < 0, then the scheme is unstable (see equation (9)). While the non-optimized second order scheme is unconditionally stable, the non-optimized third order scheme is only conditionally stable. The optimized scheme is weakly unstable, however the level has been minimized, and the lower limit was chosen to avoid instability on a typical turbomachinery problem.

Table 1. Coefficients for time derivative discretization Discretization а і a2 аз a4 2nd order 1.5 -2.0 0.5 0.0 3rd order 1.83 -3.0 1.5 0.33 2nd order optimized 1.69 -2.58 1.08 -0.19