We construct semi-hyperbolic patches of solutions, in which one family out of two families of wave characteristics start on sonic curves and end on transonic shock waves, to the two-dimensional (2D) compressible magnetohydrodynamic (MHD) equations. This type of flow patches appear frequently in transonic flow problems. In order to use the method of characteristic decomposition to construct such a flow patch, we also derive a group of characteristic decompositions for 2D self-similar MHD equations.
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Pseudosonic curve. (Ⅰ) Keldysh type; (Ⅱ) Tricomi type
A semi-hyperbolic patch
Left: small Goursat problem; right: global solution