126
3. Beyond the One-Way WaveEquation
approximation to the leading-order truncation error in the forward time difference,
In addition to its lack of accuracy, (3.37) is unstable for all 13.1 .
A stable second-order Lax-Wendroff approximation to the two-dimensional
constant-wind-speed advection equation may be written in the form
Necessary and sufficient conditions for the stability of this method are that
J.L2/3 + v 2 / 3 :s 1
(Turkel 1977). If C is abound on the magnitude of the two-dimensional wind vector and ßx = ßy = Ss, the stability condition becomes C 13.1/ßs :s i, which is
more restrictive than that for the two-dimensionalleapfrog and upstream schemes,
and much more restrictive than the stability condition for the CTU method.
The stability of the two-dirnensional Lax-Wendroff approximation can be greatly
improved using an upstream finite-difference approximation to the mixed spatial
derivative (Leonard et al. 1993). If U :::: 0 and V :::: 0, the resulting scheme is
The stability condition for this scheme is identical to that for the CTU method,
o :s J.L :s land 0 :s v :s 1 (Hong et al. 1997). If the mixed spatial derivative
is calculated in the upstream direction and the magnitude of the two-dimensional
wind vector is bounded by C , the stability condition for an isotropie mesh may be
expressed as C M / ßs s I.
Numerical solutions computed using (3.38) appear in Fig. 3.5c. The initial condition and the physical and numerical parameters are identical to those used to
obtain the two-dimensional upstream and CTU solutions. As might be expected
in problems where there is adequate numerical resolution, the amplitude error in
the second-order solution is far less than that in either first-order solution. The
leading-order dispersive error in the second-order method does, however, generate regions where cP is slightly negative. Techniques for minimizing or eliminating
these spurious negative values will be discussed in Chapter 5.
3.2.2 Systems 0/Equations in Several Dimensions
Although the stability analysis of systems of linear finite-difference equations in
several spatial dimensions is conceptually straightforward, in practice it can be
3. Beyond the One-Way WaveEquation
approximation to the leading-order truncation error in the forward time difference,
In addition to its lack of accuracy, (3.37) is unstable for all 13.1 .
A stable second-order Lax-Wendroff approximation to the two-dimensional
constant-wind-speed advection equation may be written in the form
Necessary and sufficient conditions for the stability of this method are that
J.L2/3 + v 2 / 3 :s 1
(Turkel 1977). If C is abound on the magnitude of the two-dimensional wind vector and ßx = ßy = Ss, the stability condition becomes C 13.1/ßs :s i, which is
more restrictive than that for the two-dimensionalleapfrog and upstream schemes,
and much more restrictive than the stability condition for the CTU method.
The stability of the two-dirnensional Lax-Wendroff approximation can be greatly
improved using an upstream finite-difference approximation to the mixed spatial
derivative (Leonard et al. 1993). If U :::: 0 and V :::: 0, the resulting scheme is
The stability condition for this scheme is identical to that for the CTU method,
o :s J.L :s land 0 :s v :s 1 (Hong et al. 1997). If the mixed spatial derivative
is calculated in the upstream direction and the magnitude of the two-dimensional
wind vector is bounded by C , the stability condition for an isotropie mesh may be
expressed as C M / ßs s I.
Numerical solutions computed using (3.38) appear in Fig. 3.5c. The initial condition and the physical and numerical parameters are identical to those used to
obtain the two-dimensional upstream and CTU solutions. As might be expected
in problems where there is adequate numerical resolution, the amplitude error in
the second-order solution is far less than that in either first-order solution. The
leading-order dispersive error in the second-order method does, however, generate regions where cP is slightly negative. Techniques for minimizing or eliminating
these spurious negative values will be discussed in Chapter 5.
3.2.2 Systems 0/Equations in Several Dimensions
Although the stability analysis of systems of linear finite-difference equations in
several spatial dimensions is conceptually straightforward, in practice it can be