FastPlasmaSim

Motivation

‍Acceleration of these Free-boundary Grad-Shafranov codes is of particular importance for so-called tokamak ‘flight simulators’ that are being developed to test a tokamak discharge before it is run. Such simulators should be able to simulate the entire discharge from the coil magnetization until termination and is expected to be a key component of future tokamak operations including ITER and DEMO. Ideally, these flight simulators should operate at speeds exceeding real-time to allow testing of the plasma control system in a realistic setting. Furthermore, faster-than-real-time Grad-Shafranov solvers enable routine use of optimization methods to improve tokamak operations, e.g., by optimizing poloidal field coil current evolution to maximize discharge duration or minimize structural forces.

Proposed Approach / Solution

‍Formulating the solution to the Grad-Shafranov equation involves solving a set of nonlinear equations in a high-dimensional space. The current method, a Newton schema called Jacobian-Free Newton Krylov (JFNK), addresses the challenge of Jacobian inversion costs by employing a Krylov iterative method (GMRES) and substituting Jacobian-vector products with finite differences. However, the effectiveness of this approach relies on finding good preconditioners to diminish the search dimension in the Krylov space.

We proposed an improvement of the vanilla Broyden method that leverages on the structure of the Grad-Shafranov equations, particularly the spatial localization of the plasma, resulting in sparse Jacobian. While the vanilla Broyden method constructs and maintains an approximation of the inverse Jacobian, thereby avoiding inversion costs and the necessity for effective preconditioners, its complexity remains comparable to the JFNK method. However, we achieved a reduction in computational, by maintaining a low-rank approximation of a structured decomposition of the inverse Jacobian to speed up matrix-vector products.

Impact

‍The code employing the Broyden method, which has been released in the MEQ suite, offers more than two-fold speed up in comparison to the JFNK solver (Figure 2), while keeping the convergence properties of the latter. The complexity is further reduced to linear due to the low rank approximation, which is a step towards real-time simulations and accelerates other code relying on FGS/FGE as subroutine.