It has been recognized for some time that it is necessary to go beyond the simple "resistive MHD" description of the plasma in order to get the correct quantitative results for the growth and saturation of global dissipative modes in a fusion device. The inclusion of a more complete "generalized Ohms law" and the off-diagonal terms in the ion pressure tensor introduce Whistler waves, Kinetic Alfven waves, and gyro-viscous waves, all of which are dispersive and require special numerical treatment. We have developed a new numerical approach to solving these Extended-MHD equations using a compact representation that is specifically designed to yield efficient high-order-of-accuracy, implicit solutions of a general formulation of the compressible Extended-MHD equations. The representation is based on a triangular finite element with fifth order accuracy that is constructed to have continuous derivatives across element boundaries, allowing its' use with systems of equations containing complex spatial derivative operators of up to fourth order. The magnetic and velocity fields are decomposed without loss of generality in a potential, stream function form. Subsets of the full set of six equations describing unreduced compressible extended MHD yield (1) the two variable reduced MHD equations, and (2) the four-field Fitzpatrick-Porcelli equations. Applications are presented showing the effect of "two-fluid" terms on reconnection and stability growth rates.
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