The exact flow sensitivities with repect to the high pressure for the Reimann problem

(These plots are for the nondimensional time t= 0.148.) These are plots of the derivatives of the exact solution of the Reimann problem with repect to the initial high pressure P4. We only differentiate in regions of smooth flow. We have not differentiated across the shock wave, the contact disconinuity, nor the edges of the rarefaction wave.

Again, as for the exact solution, the flow sensitivities of the Reimann problem exhibit five distinct regions:

Note that since the flow variables are discontinuous at the edges of the rarefaction wave, the sensitivities themselves are discontinuous there. We have not differentiated across the shock wave or contact discontinuity so that the delta functions that would result from such a process do not appear in these plots. At all points of discontinuity in the flow or of its derivatives, we have essentially taken one-sided derivatives of the exact flow solution; these are well-defined. Note that the even in the absence of the delta functions, the exact flow sensitivities are discontinuous at the shock wave and contact discontinuity.

A flow sensitivity along the comparison plane

The exact solution of the Reimann problem

Approximate sensitivities for the Lax-Wendroff method

Approximate sensitivities for the Godunov method

Approximate sensitivities for the Roe method.

Approximate sensitivities for the Lax-Wendroff method on the medium grid

Approximate sensitivities for the Lax-Wendroff method on the coarse grid.

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