scholarly journals Adaptive Moving Mesh Finite Volume Method Experimentation to Solving Riemann's Type Hyperbolic Problems

2021 ◽  
Vol 9 (2) ◽  
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Moving Mesh ◽  
Hyperbolic Problems ◽  
Volume Method

scholarly journals A finite volume method for two-moment cosmic-ray hydrodynamics on a moving mesh

2021 ◽  
Author(s):  
T Thomas ◽  
C Pfrommer ◽  
R Pakmor
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Source Term ◽  
Cosmic Ray ◽  
Moving Mesh ◽  
Test Problems ◽  
Integration Step ◽  
Custom Made ◽  
Anisotropic Transport ◽  
Volume Method

Abstract We present a new numerical algorithm to solve the recently derived equations of two-moment cosmic ray hydrodynamics (CRHD). The algorithm is implemented as a module in the moving mesh Arepo code. Therein, the anisotropic transport of cosmic rays (CRs) along magnetic field lines is discretised using a path-conservative finite volume method on the unstructured time-dependent Voronoi mesh of Arepo. The interaction of CRs and gyroresonant Alfvén waves is described by short-timescale source terms in the CRHD equations. We employ a custom-made semi-implicit adaptive time stepping source term integrator to accurately integrate this interaction on the small light-crossing time of the anisotropic transport step. Both the transport and the source term integration step are separated from the evolution of the magneto-hydrodynamical equations using an operator split approach. The new algorithm is tested with a variety of test problems, including shock tubes, a perpendicular magnetised discontinuity, the hydrodynamic response to a CR overpressure, CR acceleration of a warm cloud, and a CR blast wave, which demonstrate that the coupling between CR and magneto-hydrodynamics is robust and accurate. We demonstrate the numerical convergence of the presented scheme using new linear and non-linear analytic solutions.

An Essentially Nonoscillatory Spectral Deferred Correction Method for Hyperbolic Problems

2016 ◽  
Vol 13 (03) ◽  
pp. 1650017 ◽  
Author(s):  
Samet Y. Kadioglu
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Fourth Order ◽  
High Order ◽  
Computational Method ◽  
Hyperbolic Problems ◽  
Discontinuous Solutions ◽  
Order Of Accuracy ◽  
Volume Method ◽  
Essentially Nonoscillatory

We present a computational method based on the spectral deferred corrections (SDC) time integration technique and the essentially nonoscillatory (ENO) finite volume method for hyperbolic problems. The SDC technique is used to advance the solutions in time with high-order of accuracy. The ENO method is used to define high-order cell edge quantities that are then used to evaluate numerical fluxes. The coupling of the SDC method with a high-order finite volume method (piece-wise parabolic method (PPM)) is first carried out by Layton et al. [J. Comput. Phys. 194(2) (2004) 697]. Issues about this approach have been addressed and some improvements have been added to it in Kadioglu et al. [J. Comput. Math. 1(4) (2012) 303]. Here, we investigate the implications when the PPM method is replaced with the well-known ENO method. We note that the SDC-PPM method is fourth-order accurate in time and space. Therefore, we kept the order of accuracy of the ENO procedure as fourth-order in order to be able to make a consistent comparison between the two approaches (SDC-ENO versus SDC-PPM). We have tested the new SDC-ENO technique by solving smooth and nonsmooth hyperbolic problems. Our numerical results indicate that the fourth-order of accuracy in both space and time has been achieved for smooth problems. On the other hand, the new method performs very well when it is applied to nonlinear problems that involve discontinuous solutions. In other words, we have obtained highly resolved discontinuous solutions with essentially no-oscillations at or around the discontinuities.

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