TY - JOUR

T1 - High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro-dynamics

AU - Dumbser, Michael

AU - Peshkov, Ilya

AU - Romenski, Evgeniy

AU - Zanotti, Olindo

PY - 2017/11/1

Y1 - 2017/11/1

N2 - In this paper, we propose a new unified first order hyperbolic model of Newtonian continuum mechanics coupled with electro-dynamics. The model is able to describe the behavior of moving elasto-plastic dielectric solids as well as viscous and inviscid fluids in the presence of electro-magnetic fields. It is actually a very peculiar feature of the proposed PDE system that viscous fluids are treated just as a special case of elasto-plastic solids. This is achieved by introducing a strain relaxation mechanism in the evolution equations of the distortion matrix A, which in the case of purely elastic solids maps the current configuration to the reference configuration. The model also contains a hyperbolic formulation of heat conduction as well as a dissipative source term in the evolution equations for the electric field given by Ohm's law. Via formal asymptotic analysis we show that in the stiff limit, the governing first order hyperbolic PDE system with relaxation source terms tends asymptotically to the well-known viscous and resistive magnetohydrodynamics (MHD) equations. Furthermore, a rigorous derivation of the model from variational principles is presented, together with the transformation of the Euler–Lagrange differential equations associated with the underlying variational problem from Lagrangian coordinates to Eulerian coordinates in a fixed laboratory frame. The present paper hence extends the unified first order hyperbolic model of Newtonian continuum mechanics recently proposed in [110,42] to the more general case where the continuum is coupled with electro-magnetic fields. The governing PDE system is symmetric hyperbolic and satisfies the first and second principle of thermodynamics, hence it belongs to the so-called class of symmetric hyperbolic thermodynamically compatible systems (SHTC), which have been studied for the first time by Godunov in 1961 [61] and later in a series of papers by Godunov and Romenski [67,69,119]. An important feature of the proposed model is that the propagation speeds of all physical processes, including dissipative processes, are finite. The model is discretized using high order accurate ADER discontinuous Galerkin (DG) finite element schemes with a posteriori subcell finite volume limiter and using high order ADER-WENO finite volume schemes. We show numerical test problems that explore a rather large parameter space of the model ranging from ideal MHD, viscous and resistive MHD over pure electro-dynamics to moving dielectric elastic solids in a magnetic field.

AB - In this paper, we propose a new unified first order hyperbolic model of Newtonian continuum mechanics coupled with electro-dynamics. The model is able to describe the behavior of moving elasto-plastic dielectric solids as well as viscous and inviscid fluids in the presence of electro-magnetic fields. It is actually a very peculiar feature of the proposed PDE system that viscous fluids are treated just as a special case of elasto-plastic solids. This is achieved by introducing a strain relaxation mechanism in the evolution equations of the distortion matrix A, which in the case of purely elastic solids maps the current configuration to the reference configuration. The model also contains a hyperbolic formulation of heat conduction as well as a dissipative source term in the evolution equations for the electric field given by Ohm's law. Via formal asymptotic analysis we show that in the stiff limit, the governing first order hyperbolic PDE system with relaxation source terms tends asymptotically to the well-known viscous and resistive magnetohydrodynamics (MHD) equations. Furthermore, a rigorous derivation of the model from variational principles is presented, together with the transformation of the Euler–Lagrange differential equations associated with the underlying variational problem from Lagrangian coordinates to Eulerian coordinates in a fixed laboratory frame. The present paper hence extends the unified first order hyperbolic model of Newtonian continuum mechanics recently proposed in [110,42] to the more general case where the continuum is coupled with electro-magnetic fields. The governing PDE system is symmetric hyperbolic and satisfies the first and second principle of thermodynamics, hence it belongs to the so-called class of symmetric hyperbolic thermodynamically compatible systems (SHTC), which have been studied for the first time by Godunov in 1961 [61] and later in a series of papers by Godunov and Romenski [67,69,119]. An important feature of the proposed model is that the propagation speeds of all physical processes, including dissipative processes, are finite. The model is discretized using high order accurate ADER discontinuous Galerkin (DG) finite element schemes with a posteriori subcell finite volume limiter and using high order ADER-WENO finite volume schemes. We show numerical test problems that explore a rather large parameter space of the model ranging from ideal MHD, viscous and resistive MHD over pure electro-dynamics to moving dielectric elastic solids in a magnetic field.

KW - Arbitrary high-order ADER Discontinuous Galerkin schemes

KW - Finite signal speeds of all physical processes

KW - Nonlinear hyperelasticity

KW - Path-conservative methods and stiff source terms

KW - Symmetric hyperbolic thermodynamically compatible systems (SHTC)

KW - Unified first order hyperbolic model of continuum physics (fluid mechanics, solid mechanics, electro-dynamics)

KW - Galerkin schemes

KW - GENERALIZED RIEMANN PROBLEM

KW - TANG VORTEX SYSTEM

KW - ADAPTIVE MESH REFINEMENT

KW - DISCONTINUOUS GALERKIN METHOD

KW - DIFFUSION-REACTION EQUATIONS

KW - Arbitrary high-order ADER Discontinuous

KW - HIGH-VELOCITY IMPACT

KW - KELVIN-HELMHOLTZ INSTABILITY

KW - STIFF RELAXATION TERMS

KW - CONSERVATION-LAWS

KW - FINITE-VOLUME SCHEMES

UR - http://www.scopus.com/inward/record.url?scp=85026425019&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2017.07.020

DO - 10.1016/j.jcp.2017.07.020

M3 - Article

AN - SCOPUS:85026425019

VL - 348

SP - 298

EP - 342

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -