Development of kinetic schemes for the numerical solutions of the two-dimensional Euler and the ideal magnetohydrodynamics equations

The focus of this dissertation is on the development and application of kinetic schemes for the numerical solution of compressible Euler and ideal magnetohydrodynamics (MHD) equations. Kinetic schemes for the Euler and Navier-Stokes equations are derived from the Boltzmann equation by employing the so-called “moment method strategy”; the Boltzmann equation is “upwind” discretized and then the moments of the discretized Boltzmann equation are taken with a collision invariant vector and the appropriate distribution function to obtain the numerical scheme for the continuum equations.; Upwind discretization of the Boltzmann equation based on the sign of the molecular velocity leads to the well-known Kinetic Flux-Vector Splitting (KFVS) algorithm. However, if the molecular velocity is expressed as = + where is the fluid velocity and is the thermal velocity, and the Boltzmann equation is upwind discretized depending upon the sign of both and , the “moment method strategy” leads to the so-called Kinetic Wave/Particle Splitting (KWPS) algorithm. In the literature, both of these algorithms (KFVS and KWPS) have been developed for first-order accurate spatial discretization with explicit time-stepping for the Euler equations, Navier-Stokes equations, Burnett equations, and ideal MHD equations.; In this dissertation, initially the first-order accurate time- explicit KFVS and KWPS algorithms are developed for the Euler and ideal MHD equations. Then the first-order accurate time-implicit KFVS and KWPS algorithms are derived for both the Euler and ideal MHD equations. It should be noted that the implicit kinetic schemes have never been formulated in the literature before; these are derived for the first time in this dissertation. The derivations are presented in the 3-D generalized coordinate system. However, the numerical validations and applications are presented only for the 2-D cases.; Both explicit and implicit kinetic schemes are validated and compared for their accuracy, stability, and efficiency for both the steady state and time-accurate calculations by computing a number of test cases for the Euler and ideal MHD equations. The implicit kinetic schemes are found to be computationally more efficient compared to the explicit schemes for achieving the same level of accuracy.; The dissertation makes a fundamental contribution to the development and application of implicit kinetic schemes.