A numerical method for the calculation of the magnetic diffusion equation in wire array z-pinch

Abstract

Wire-array Z pinch has been proved to be the most powerful X-ray sources in laboratory with crucial applications in many fields. Much effort has been devoted for better understanding of its physical process via the approaches of numerical simulation as well as experimental investigation. As is well known, the dynamics of plasma is greatly influenced by the magnetic field, hence the numerical calculation of magnetic field plays a crucial role in the simulation of wire-array z-pinch. In this work, an algorithm for calculating the evolution of magnetic field, which is consistent with corresponding numerical approaches of MHD simulation, is proposed. Above all, an operator split method is used to break the magnetic diffusion equation into two parts, representing the “frozen” effect and the “diffusion” effect, respectively. For the calculation of the “frozen” part, the principal difficulty is that Maxwell's equations require the magnetic field to satisfy the divergence-free constraint all the time. The traditional approach to circumvent this difficulty is to involve the magnetic vector potential. However, this method could reduce the order of accuracy of Lorentz force which requires taking a second derivative of the vector potential. In our work, we implement a constrained transport method1 which directly incorporates the divergence-free constraint into the finite-difference equation. The “diffusion” part is also broken into parts with each part representing one dimension. After that, they are central differenced and solved implicitly by tridiagonal matrix method (TDMA), respectively. The result shows good agreement with analytical solutions.