A Non-iterative Hyperbolic, First-order Conservation Law Approach to Divergence-free Solutions to Maxwell’s Equations Richard J. Thompson 1 and Trevor Moeller 2 1 Applied Mathematician Boeing Research & Technology Mathematics and Applied Analysis Huntsville, AL Assistant Professor University of Tennessee Space Institute Dept. of Mechanical, Aerospace & Biomedical Engineering Tullahoma, TN AIAA AVIATION 2015 Dallas, TX; June 22, 2015 Technical Paper AIAA-2015-

Overview Previous work used finite volume methods to solve the coupled fluid and Maxwell equations for MHD and MHD-like plasma simulations. Solution of the Maxwell equations in the finite volume solver required iteration to converge the solution to a divergence-free (physical) solution. However, the iteration required to converge the solution to the divergence-free solution often was slow, and had to be performed every timestep. This immensely slowed the simulation process down. This investigation resulted from trying to find a way to determine a divergence- free solution without iteration, but requiring it to fit in the current first-order, hyperbolic finite volume formulation being used in our solver. We succeeded in finding a formulation that seems to work (though our analysis of the uncertainty is still ongoing), by rewriting the potential formulation of the Maxwell equations in a 1 st -order, hyperbolic form. This is the new method being presented in this work.

Motivation Hypersonic flight/propulsion Electrohydrodynamic flow control Space physics Blackout communications in flight Atmospheric communications effects Electric space propulsion EM wave simulation

Maxwell’s Equations Maxwell’s equations for E- and B-field: The two divergence equations are divergence constraints. They must be satisfied for the solution to be a physical solution. However, it is difficult numerically to satisfy these two divergence equations to determine the divergence- free solution. Typically, numerically, one must either clean the divergence every timestep to keep the solution constrained, or somehow construct a divergence-free numerical method. The former typically requires a lot of iterations every timestep, and can be very expensive. The latter usually requires a very complicated construction of the solution every timestep, and can present implementation challenges.

Maxwell’s Equations Another approach is to construct a potential formulation for Maxwell’s equations, and solve 4 2 nd -order equations instead for a vector and scalar potential: The electromagnetic fields can be calculated as derivatives of the potentials, once they are known: This approach automatically satisfies the divergence equations, and thus the solution to this system represents the divergence-free solution that obeys the divergence constraints. However, the potential equations are 2 nd -order, and not directly amenable to the finite volume method. This can also complicate how they are solved on complex geometries. In this investigation, we attempted to construct a 1 st -order method amenable directly to the finite volume method to solve the potential formulation. This can be done directly (without iteration), and may be directly implemented in our finite volume solver with little modification.

Problem: We want to solve Maxwell’s equations for wave propagation or MHD systems using the finite volume method, since it fits into our current software, but…. …solving Maxwell’s equations using either the perfectly hyperbolic equations or Brackbill-Barnes formulation results in very slow convergence of results to the divergence-free solution. The new method presented in this work was motivated by wanting to fit the Maxwell solver into a first-order, hyperbolic finite volume formulation, but doing it without the slow convergence to a divergence-free solution.

Solution We constructed a new method to try to resolve the divergence-free solution without iteration, as a first-order system of equations. The new method solves the potential formulation for Maxwell’s equations. However, it does not solve for the vector/scalar potentials. We take the derivatives of the vector and scalar potential as the new unknown dynamical quantities: We instead solve a system of 16 1 st -order equations for R, S, T, and U Electromagnetic fields can then be calculated algebraically:

New Approach First 4 eqns are just potential equations in terms of R, S, T, and U: These are hyperbolic, first-order equations that are directly amenable to fitting into the finite volume method without modification, or iteration However, these are 4 equations for 16 unknowns. We must find more equations to determine a unique solution.

New Approach The remaining equations come from derivative identities – switching the order of differentiation cannot matter. Hence: In terms of R, S, T and U, these become hyperbolic constraint equations: This adds 12 equations to the system, giving a unique 16x16 system.

New Approach – the mathematical problem Thus, the collected 16 equations are: For 16 unknown derivative fields: From which the EM fields are: This is a linear system of 16x16 equations with constant eigenvalues 0, +/-c, and with a very simple eigenstructure. We tested both Roe and Rusanov numerical flux methods, and both gave nearly identical results.

New Approach – some notes Some important notes of the new system are in order. The system is composed of hyperbolic, 1 st -order equations, so fits directly into the finite volume method. It is also linear, so eigenstructure is very simple (constant eigenvalues: 0, +/- speed of light) We developed both Roe and Rusanov numerical flux approximations to this system: they turn out here to be nearly identical in formulation. The system was constructed from the potential equations, and so satisfies the divergence-free solution. No iteration is needed to solve; a direct solve gives the divergence-free solution. Note that U is not a symmetric matrix. Note that the Lorentz gauge becomes: You can use this algebraic Lorentz gauge relation to solve R algebraically, reducing the system to a 15x15 system. We implemented both the 16x16 and 15x15, and both gave identical results.

Results Currently: 1D and 2D tests, including both wave propagation and MHD simulations, have been successfully run The results of these tests look very promising, and matches expected solution results In the future: We will try applying the new method to other various simulation tests We want to quantify the improvement, stability and robustness of the method Will perform a quantitative assessment of the uncertainty and divergence error

Results – 1D 1D wave propagation test Very simple to implement and simulate Allows for a quantitative assessment of the solver’s physical correctness Computational domain: 1 meter length in X direction Boundary conditions: Neumann on both sides Initial conditions: Expected results: The initial discontinuity in Bz will split, two co-propagating waves result The waves must move at the physical speed of light A corresponding electric field in Y should be induced by the presence of the magnetic wave

Results – 1D

Results – 2D

MHD vortex initial conditions: Gamma is 1.4

Results – 2D Density contour plotPressure contour plot

Results – 2D Velocity field vector plotMagnetic field vector plot

Results – 2D

MHD rotor initial conditions: (Gamma = 1.4)

Results – 2D Density contour plotPressure contour plot

Results – 2D Velocity field vector plotMagnetic field vector plot

Conclusions Motivated by slow convergence of magnetic fields because of divergence, we developed a new method solving the potential equations in a first-order, hyperbolic form. Only four of the first-order equations are provided via Maxwell equations. The remainder of equations to make the system unique are constructed from derivative identities. 1D and 2D tests show promising results that are expected for wave propagation of electromagnetic fields, and MHD field evolution In the near future, we will characterize/analyze/quantify the solution uncertainty and divergence error of new solution method