On the Divergence‐free Condition and Conservation Laws in Numerical Simulations for Supersonic Magnetohydrodynamical Flows

1998 ◽  
Vol 494 (1) ◽  
pp. 317-335 ◽  
Author(s):  
Wenlong Dai ◽  
Paul R. Woodward
Keyword(s):  
Conservation Laws ◽  
Numerical Simulations ◽  
Free Condition ◽  
Divergence Free

scholarly journals On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 878
Author(s):  
Alexei Cheviakov ◽  
Denys Dutykh ◽  
Aidar Assylbekuly
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Local Symmetry ◽  
Symmetry And Conservation Laws ◽  
Higher Order ◽  
Special Equation ◽  
Long Wave ◽  
Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

scholarly journals Topology optimization for three-dimensional electromagnetic waves using an edge element-based finite-element method

2016 ◽  
Vol 472 (2189) ◽  
pp. 20150835 ◽  
Author(s):  
Yongbo Deng ◽  
Jan G. Korvink
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Topology Optimization ◽  
Electromagnetic Waves ◽  
Three Dimensional ◽  
Free Condition ◽  
Edge Element ◽  
Divergence Free ◽  
The Magnetic Field ◽  
Element Method

This paper develops a topology optimization procedure for three-dimensional electromagnetic waves with an edge element-based finite-element method. In contrast to the two-dimensional case, three-dimensional electromagnetic waves must include an additional divergence-free condition for the field variables. The edge element-based finite-element method is used to both discretize the wave equations and enforce the divergence-free condition. For wave propagation described in terms of the magnetic field in the widely used class of non-magnetic materials, the divergence-free condition is imposed on the magnetic field. This naturally leads to a nodal topology optimization method. When wave propagation is described using the electric field, the divergence-free condition must be imposed on the electric displacement. In this case, the material in the design domain is assumed to be piecewise homogeneous to impose the divergence-free condition on the electric field. This results in an element-wise topology optimization algorithm. The topology optimization problems are regularized using a Helmholtz filter and a threshold projection method and are analysed using a continuous adjoint method. In order to ensure the applicability of the filter in the element-wise topology optimization version, a regularization method is presented to project the nodal into an element-wise physical density variable.

scholarly journals Divergence-free condition in transport simulation

2016 ◽  
Vol 344 (9) ◽  
pp. 642-648 ◽  
Author(s):  
Adrien Berchet ◽  
Anthony Beaudoin ◽  
Serge Huberson
Keyword(s):  
Free Condition ◽  
Transport Simulation ◽  
Divergence Free

scholarly journals On the inadmissibility of non-evolutionary shocks

2001 ◽  
Vol 65 (1) ◽  
pp. 29-58 ◽  
Author(s):  
S. A. E. G. FALLE ◽  
S. S. KOMISSAROV
Keyword(s):  
Conservation Laws ◽  
Numerical Simulations ◽  
Classical Theory ◽  
Existence And Uniqueness ◽  
Numerical Solutions ◽  
Entropy Function ◽  
Numerical Schemes ◽  
Discontinuous Solutions ◽  
Free System

In recent years, numerical solutions of the equations of compressible magnetohydrodynamic (MHD) flows have been found to contain intermediate shocks for certain kinds of problems. Since these results would seem to be in conflict with the classical theory of MHD shocks, they have stimulated attempts to reexamine various aspects of this theory, in particular the role of dissipation. In this paper, we study the general relationship between the evolutionary conditions for discontinuous solutions of the dissipation-free system and the existence and uniqueness of steady dissipative shock structures for systems of quasilinear conservation laws with a concave entropy function. Our results confirm the classical theory. We also show that the appearance of intermediate shocks in numerical simulations can be understood in terms of the properties of the equations of planar MHD, for which some of these shocks turn out to be evolutionary. Finally, we discuss ways in which numerical schemes can be modified in order to avoid the appearance of intermediate shocks in simulations with such symmetry.

scholarly journals High Order Hierarchical Divergence-Free Constrained Transport H(div) Finite Element Method for Magnetic Induction Equation

2017 ◽  
Vol 10 (2) ◽  
pp. 243-254 ◽  
Author(s):  
Wei Cai ◽  
Jun Hu ◽  
Shangyou Zhang
Keyword(s):  
Magnetic Field ◽  
Finite Element Method ◽  
Finite Element ◽  
Magnetic Induction ◽  
High Order ◽  
Free Condition ◽  
Induction Equation ◽  
Divergence Free ◽  
Magnetic Induction Equation ◽  
Element Method

AbstractIn this paper, we propose to use the interior functions of an hierarchical basis for high order BDMp elements to enforce the divergence-free condition of a magnetic field B approximated by the H(div)BDMp basis. The resulting constrained finite element method can be used to solve magnetic induction equation in MHD equations. The proposed procedure is based on the fact that the scalar (p–1)-th order polynomial space on each element can be decomposed as an orthogonal sum of the subspace defined by the divergence of the interior functions of the p-th order BDMp basis and the constant function. Therefore, the interior functions can be used to remove element-wise all higher order terms except the constant in the divergence error of the finite element solution of the B-field. The constant terms from each element can be then easily corrected using a first order H(div) basis globally. Numerical results for a 3-D magnetic induction equation show the effectiveness of the proposed method in enforcing divergence-free condition of the magnetic field.

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