Self-adaptive moving mesh schemes for short pulse type equations and their Lax pairs

Integrable self-adaptive moving mesh schemes for short pulse type equations (the short pulse equation, the coupled short pulse equation, and the complex short pulse equation) are investigated. Two systematic methods, one is based on bilinear equations and another is based on Lax pairs, are shown. Self-adaptive moving mesh schemes consist of two semi-discrete equations in which the time is continuous and the space is discrete. In self-adaptive moving mesh schemes, one of two equations is an evolution equation of mesh intervals which is deeply related to a discrete analogue of a reciprocal (hodograph) transformation. An evolution equations of mesh intervals is a discrete analogue of a conservation law of an original equation, and a set of mesh intervals corresponds to a conserved density which play an important role in generation of adaptive moving mesh. Lax pairs of self-adaptive moving mesh schemes for short pulse type equations are obtained by discretization of Lax pairs of short pulse type equations, thus the existence of Lax pairs guarantees the integrability of self-adaptive moving mesh schemes for short pulse type equations. It is also shown that self-adaptive moving mesh schemes for short pulse type equations provide good numerical results by using standard time-marching methods such as the improved Euler's method.Comment: 13 pages, 6 figures, To be appeared in Journal of Math-for-Industr

A moving mesh method for one-dimensional hyperbolic conservation laws

We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work

Non-ideal magnetohydrodynamics on a moving mesh

In certain astrophysical systems the commonly employed ideal magnetohydrodynamics (MHD) approximation breaks down. Here, we introduce novel explicit and implicit numerical schemes of ohmic resistivity terms in the moving-mesh code AREPO. We include these non-ideal terms for two MHD techniques: the Powell 8-wave formalism and a constrained transport scheme, which evolves the cell-centred magnetic vector potential. We test our implementation against problems of increasing complexity, such as one- and two-dimensional diffusion problems, and the evolution of progressive and stationary Alfv\'en waves. On these test problems, our implementation recovers the analytic solutions to second-order accuracy. As first applications, we investigate the tearing instability in magnetized plasmas and the gravitational collapse of a rotating magnetized gas cloud. In both systems, resistivity plays a key role. In the former case, it allows for the development of the tearing instability through reconnection of the magnetic field lines. In the latter, the adopted (constant) value of ohmic resistivity has an impact on both the gas distribution around the emerging protostar and the mass loading of magnetically driven outflows. Our new non-ideal MHD implementation opens up the possibility to study magneto-hydrodynamical systems on a moving mesh beyond the ideal MHD approximation.Comment: 18 pages, 11 figures, accepted for publication in MNRAS. Revisions to match the accepted versio