Dispersion-Relation-Preserving (DRP) Scheme

Now, consider solving the linearized Euler equation (5.1) computationally. It would be advantageous to first rewrite the equation in the following form:

so that the time derivative is by itself. Eq. (5.26) may be discretized by using the 7-point optimized finite difference scheme and Eq. (5.27) by using the four-level optimized time marching scheme. The fully discretized equations are as follows:

where l, m, and n are the x, у, and t indices and Ax, Ay, and At are the mesh sizes. Eqs. (5.28) and (5.29) form a self-contained time marching scheme. If the values of U are known at time level n, n – 1, n – 2, and n – 3, then Kl()m (i = n, n – 1, n – 2, n – 3) may be calculated according to Eq. (5.28). By Eq. (5.29) the values of U(”,+1)
may be determined. With U("„+1) known, Kl”,+1) may be calculated by Eq. (5.28) and marched to the next time level, n + 2, by using Eq. (5.29). The process is repeated until the desired time level is reached.

The Euler equations are first-order in time, so the initial conditions are as follows:

t = 0 U(X, y, 0) = Uinitial(X, y). (530)

This initial condition is sufficient for the single time step method. However, it will be shown that the appropriate starting condition for a multilevel time marching scheme is

U(0i = U initial’ = 0 fornegative П. (5.31)

At this time, the two pertinent questions to ask are, “How good is the computation scheme (5.28), (5.29), and (5.31)?”; “Do they provide entropy, vorticity, and acoustic wave solutions that have the same wave propagation characteristics as those of the original linearized Euler equations?” The answer to the second question is that the marching scheme does guarantee that the long waves (X > 7Ax) of the numerical solution would have almost identical wave propagation properties as the corresponding waves of the linearized Euler equations. The answer to the first question is that the solution would be numerically accurate. This is because it will be shown that the finite difference algorithm has formally identical dispersion relations as the original linearized Euler equations. For this reason, the algorithm is referred to as the DRP scheme (Tam and Webb, 1993).

To find the dispersion relations of the time marching scheme, it is necessary first to generalize the discrete finite difference equations to finite difference equa­tions with continuous variables. The complete initial value problem consists of the following equations and initial conditions.

The following Shifting Theorems for Laplace transforms (see Appendix A) are useful.

A > 0

2^ j f (t + A)eimtdt = e-imA f(v) – I 2- I f (t)eimtdt I e-imA

The Fourier-Laplace transforms of Eqs. (5.32) and (5.33) with initial condition (5.34) are

K = —i aE – i в IF + H (5.36)

– i mU = K + – П-Uinitial. (5.37)

2П M

Upon eliminating IK, the two equations may be combined to yield the following:

AU = G, (5.38)

where A is the same as A of Eq. (5.4) provided a, в, and м are replaced by a, в, M, and G = i[H + (MUinitial/M2n)].

A closer examination of Eq. (5.38) indicates that it is the same as Eq. (5.4) except with the replacement of a, в, м by a, в, m. In the long-wave range, it was shown that

a — a, в — в, and m — m.

This means that the Fourier-Laplace transform of the DRP scheme is formally the same as the Fourier-Laplace transform of the original partial differential equations. This guarantees that the two systems would have formally the same dispersion relations. The solutions would automatically have nearly the same properties and the same asymptotic solutions. In other words, for long waves, the numerical solution is identical to that of the solution of the partial differential equations to the extent a — a, в — в, and M — m.