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from Part I - 1-D MHD in Ten Weeks
Published online by Cambridge University Press: 05 June 2025
In this chapter, everything is brought together to solve the MHD Riemann problem, the most general 1-D MHD problem one can solve semi-analytically. Non-linear waves are introduced in which the 1-D primitive equations are neither steady-state nor linearised. The fast and slow eigenkets are evaluated and their normalisation to account for the not-strictly hyperbolic nature of MHD is emphasised. A method to determine profiles of the primitive variables across slow and fast rarefaction fans is described, including Euler, switch-on, and switch-off fans. A strategy for solving the MHD Riemann problem follows, including use of a multi-variate secant root finder, sixth- order Runge–Kutta, and inverting a 5 × 5 Jacobian matrix with emphasis on characteristic degeneracy and matrix singularity. The chapter concludes with an explicit algorithm for an MHD Riemann solver including numerous examples using a solver developed by the author.
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