8 research outputs found
Higher order finite difference schemes for the magnetic induction equations
We describe high order accurate and stable finite difference schemes for the
initial-boundary value problem associated with the magnetic induction
equations. These equations model the evolution of a magnetic field due to a
given velocity field. The finite difference schemes are based on Summation by
Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation
Term (SAT) technique for imposing boundary conditions. We present various
numerical experiments that demonstrate both the stability as well as high order
of accuracy of the schemes.Comment: 20 page
Structure preserving numerical methods for the ideal compressible MHD system
We introduce a novel structure-preserving method in order to approximate the
compressible ideal Magnetohydrodynamics (MHD) equations. This technique
addresses the MHD equations using a non-divergence formulation, where the
contributions of the magnetic field to the momentum and total mechanical energy
are treated as source terms. Our approach uses the Marchuk-Strang splitting
technique and involves three distinct components: a compressible Euler solver,
a source-system solver, and an update procedure for the total mechanical
energy. The scheme allows for significant freedom on the choice of Euler's
equation solver, while the magnetic field is discretized using a
curl-conforming finite element space, yielding exact preservation of the
involution constraints. We prove that the method preserves invariant domain
properties, including positivity of density, positivity of internal energy, and
the minimum principle of the specific entropy. If the scheme used to solve
Euler's equation conserves total energy, then the resulting MHD scheme can be
proven to preserve total energy. Similarly, if the scheme used to solve Euler's
equation is entropy-stable, then the resulting MHD scheme is entropy stable as
well. In our approach, the CFL condition does not depend on magnetosonic
wave-speeds, but only on the usual maximum wave speed from Euler's system. To
validate the effectiveness of our method, we solve a variety of ideal MHD
problems, showing that the method is capable of delivering high-order accuracy
in space for smooth problems, while also offering unconditional robustness in
the shock hydrodynamics regime as well
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An upwind vertex centred finite volume algorithm for nearly and truly incompressible explicit fast solid dynamic applications: Total and Updated Lagrangian formulations
This paper presents an explicit vertex centred finite volume method for the solution of fast transient isothermal large strain solid dynamics via a system of first order hyperbolic conservation laws. Building upon previous work developed by the authors, in the context of alternative numerical discretisations, this paper explores the use of a series of enhancements (both from the formulation and numerical standpoints) in order to explore some limiting scenarios, such as the consideration of near and true incompressibility. Both Total and Updated Lagrangian formulations are presented and compared at the discrete level, where very small differences between both descriptions are observed due to the excellent discrete satisfaction of the involutions. In addition, a matrix-free tailor-made artificial compressibility algorithm is discussed and combined with an angular momentum projection algorithm. A wide spectrum of numerical examples is thoroughly examined. The scheme shows excellent (stable, consistent and accurate) behaviour, in comparison with other methodologies, in compressible, nearly incompressible and truly incompressible bending dominated scenarios, yielding equal second order of convergence for velocities, deviatoric and volumetric components of the stress