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Abstract Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations (PDEs). The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions. A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems. Hence, they are easy to be applied to a general hyperbolic system. To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains, inverse Lax-Wendroff (ILW) procedures were developed as a very effective approach in the literature. In this paper, we combine a fifth-order fixed-point fast sweeping WENO method with an ILW procedure to solve steady-state solution of hyperbolic conservation laws on complex computing regions. Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids. Numerical results show high-order accuracy and good performance of the method. Furthermore, the method is compared with the popular third-order total variation diminishing Runge-Kutta (TVD-RK3) time-marching method for steady-state computations. Numerical examples show that for most of examples, the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.
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A fast local embedded boundary method suitable for high power electromagnetic sources
High power sources of electromagnetic energy often require complicated structures to support electromagnetic modes and shape electromagnetic fields to maximize the coupling of the field energy to intense relativistic electron beams. Geometric fidelity is critical to the accurate simulation of these High Power Electromagnetic (HPEM) sources. Here, we present a fast and geometrically flexible approach to calculate the solution to Maxwell’s equations in vector potential form under the Lorenz gauge. The scheme is an implicit, linear-time, high-order, A-stable method that is based on the method of lines transpose (MOLT). As presented, the method is fourth order in time and second order in space, but the A-stable formulation could be extended to both high order in time and space. An O(n) fast convolution is employed for space-integration. The main focus of this work is to develop an approach to impose perfectly electrically conducting (PEC) boundary conditions in MOLT by extending our past work on embedded boundary methods. As the method is A-stable, it does not suffer from small time step limitations that are found in explicit finite difference time domain methods when using either embedded boundary or cut-cell methods to capture geometry. This is a major advance for the simulation of HPEM devices. While there is no conceptual limitation to develop this in 3D, our initial work has centered on 2D. The extension to 3D requires validation that the proposed fixed point iteration will converge and is the subject of our follow-up work. The eventual goal is to combine this method with particle methods for the simulations of plasma. In the current work, the scheme is evaluated for EM wave propagation within an object that is bounded by PEC. The consistency and performance of the scheme are confirmed using the ping test and frequency mode analysis for rotated square cavities—a standard test in the HPEM community. We then demonstrate the diffraction Q value test and the use of this method for simulating an A6 magnetron. The ability to handle both PEC and open boundaries in a standard device test problem, such as the A6, gives confidence on the robustness of this new method.
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- Award ID(s):
- 1912183
- PAR ID:
- 10490924
- Publisher / Repository:
- AIP Publishing
- Date Published:
- Journal Name:
- AIP Advances
- Volume:
- 10
- Issue:
- 11
- ISSN:
- 2158-3226
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0019210
