We present a new family of high-resolution, nonoscillatory semidiscrete central schemes for the approximate solution of the ideal magnetohydrodynamics (MHD) equations. This is the second part of our work, where we are passing from the fully discrete staggered schemes in [J. Balbás, E. Tadmor, and C.-C. Wu, J. Comput. Phys., 201 (2004), pp. 261-285] to the semidiscrete formulation advocated in [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241-282]. This semidiscrete formulation retains the simplicity of fully discrete central schemes while enhancing efficiency and adding versatility. The semidiscrete algorithm offers a wider range of options to implement its two key steps: nonoscillatory reconstruction of point values followed by the evolution of the corresponding point valued fluxes. We present the solution of several prototype MHD problems. Solutions of one-dimensional Brio-Wu shock-tube problems and the two-dimensional Kelvin-Helmholtz instability, Orszag-Tang vortex system, and the disruption of a high density cloud by a strong shock are carried out using third-and fourth-order central schemes based on the central WENO reconstructions. These results complement those presented in our earlier work and confirm the remarkable versatility and simplicity of central schemes as black-box, Jacobian-free MHD solvers. Furthermore, our numerical experiments demonstrate that this family of semidiscrete central schemes preserves the ∇ · B = 0-constraint within machine round-off error; happily, no constrained-transport enforcement is needed.

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Abstract. We present a new semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steady-states of rest (i.e., it is wellbalanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical algorithm. Mathematics Subject Classification. 65M99, 35L65. The shallow-water modelWe consider the shallow water equations along channels with non-uniform rectangular cross sections and bottom topography. The model describes flows that are nearly horizontal and can be obtained by averaging the Euler equations over the channel cross section [7], resulting in the balance lawwhere h and σ(x) are, respectively, the height of the fluid above the bottom of the channel, and the channel breadth, A = σh is the wet cross-section, Q = Au is the discharge, with u denoting the (depth average) fluid velocity, B(x) describes the bottom topography of the channel, and g is the acceleration of gravity (see Fig. 1).Keywords and phrases. Hyperbolic systems of conservation and balance laws, semi-discrete schemes, Saint-Venant system of shallow water equations, non-oscillatory reconstructions, channels with irregular geometry.

C Ce en nt te er r f fo or r S Sc ci ie en nt ti if fi ic c C Co om mp pu ut ta at ti io on n A An nd d M Ma at th he em ma at ti ic ca al l M Mo od de el li in ng g SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution unlimited SUPPLEMENTARY NOTESThe original document contains color images. AbstractIn this paper we utilize a family of high-resolution, non-oscillatory central schemes for the approximate solution of the equations of ideal magnetohydrodynamics (MHD) in one-and two-space dimensions. We present several prototype problems. Solutions of one-dimensional shock-tube problems is carried out using second-and third-order central schemes [19,18], and we use the second-order central scheme [11] which is adapted for the solution of the two-dimensional Kelvin-Helmholtz and Orszag-Tang problems. A qualitative comparison reveals an excellent agreement with previous results based on upwind schemes. Central schemes, however, require little knowledge about the eigen-structure of the problem -in fact, we even avoid an explicit evaluation of the corresponding Jacobians, while at the same time they eliminate the need for dimensional splitting. The one-and two-dimensional computations reported in this paper demonstrate the remarkable versatility of central schemes as black-box, Jacobian-free MHD solvers.

We present a new high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. In addition to these, a modification in the numerical flux and the estimate of the speed of propagation, the scheme incorporates the ability to detect and resolve partially wet regions, i.e., wet-dry states. Along with a detailed description of the scheme and proofs of its properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.