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Extra info for Hyperbolic Partial Differential Equations. Modern Applied Mathematics and Computer Science
These LCI forms preserve the conservative form of the differencing scheme; thus they are applicable to unsteady as well as steady-state calculations. The main drawback is that the resulting scheme may no longer be unconditionally TVD for 6 = 1. There is no proof to show that the LCI form is still unconditionally TVD. 13) is not unconditionally TVD. This shortcoming of the LCI form appears to be disturbing at first. In practice, however, unconditional stability arising from linear stability analysis rarely carries over to nonlinear system cases with discontinuous solutions.
C. rnl i 4|ijiiii iii||i||ij'^ ! 0 Fig. 2. 25 t / \ if c* u p if i^. 0 Fig. 3. 25. 0 Fig. 4. 0. 5 Fig. 5. 0 Fig. 6. Pressure coefficient and Mach contours for the NACA0012 airfoil with M,. 0. 0 Fig. 7. 25. 428 H. C. 0 Fig. 8. 0. employed by FL052R and ARC2D. Thus, the advantage of the current scheme over FL052R and ARC2D is that the unfavorable condition in computations which arises from the high aspect ratio of grid spacing can be avoided. Moreover, the present method requires less storage and lower operation count than other TVD schemes.
For computational efficiency, consider the following solution strategies. First, linearize the implicit operator in several ways as in the scalar case. Then construct an alternating direction implicit (ADI) form for the linearized implicit algorithms. 4) as the corrector step (with the numerical flux values evaluated at the predicted solution). The predictor-corrector method is proposed mainly because of its use (a) as a vehicle to compensate for the simplification of the various linearized implicit operators, and (b) for the possible improvement in convergence rate for steady-state applications.