Culham Center for Fusion Energy (CCFE)

MHD equilibrium is an important topic for fusion (and other MHD applications). A tokamak, in principle, is a toroidally symmetric fusion device and so MHD equilibrium can be reduced to solving the time independent MHD equations in axisymmetry. This produces the Grad-Shafranov equation (a two dimensional, nonlinear PDE) which has been solved using various techniques in the fusion community including finite difference, finite elements and spectral methods. A similar PDE exists if there is a plasma column with helical symmetry. Non-axisymmetric plasmas do occur in tokamaks as a result of instabilities and applied fields. However, if there is no symmetry angle there is no PDE to be solved. The current workhorse for finding non-axisymmetric equilibria uses energy minimization to find the equilibrium. New approaches to this problem that can use state of the art techniques are desirable. The speaker has formulated a coupled set of PDEs for the non-axisymmetric MHD equilibrium problem assuming that flux surfaces are nested (i.e. there are no magnetic islands) and has written this in weak form to use finite element method to solve the equations. The questions are around whether there is an optimal way to try to formulate the problem for FEM and to couple the equations, what sort of elements to use, if other solution techniques would be better suited and so on.

Parallelizing the time domain in numerical simulations is non-intuitive, but has been proven to be possible using various algorithms like parareal, PFASST and RIDC. Temporal parallelizations adds an entire new dimension to parallelize and significantly enhances use of super computing resources. Exploiting this technique serves as a big step towards exascale computation.

Starting with relatively simple problems, the parareal algorithm (Lions et al, A ''parareal'' in time discretization of PDE's, 2001) has been successfully applied to various complex simulations in the last few years (Samaddar et al, Parallelization in time of numerical simulations of fully-developed plasma turbulence using the parareal algorithm, 2010). The algorithm involves a predictor-corrector technique.

Numerical studies of the edge of magnetically confined, fusion plasma are an extremely challenging task. The complexity of the physics in this regime is particularly increased due to the presence of neutrals as well as the interaction of the plasma with the wall. These simulations are extremely computationally intensive but are key to rapidly achieving thermonuclear breakeven on ITER-like machines.

The SOLPS code package (Schneider et al, Plasma Edge Physics with B2‐Eirene, 2006) is widely used in the fusion community and has been used to design the ITER divertor. A reduction of the wallclock time for this code has been a long standing goal and recent studies have shown that a computational speed-up greater than 10 is possible for SOLPS (Samaddar et al, Greater than 10x Acceleration of fusion plasma edge simulations using the Parareal algorithm, 2014), which is highly significant for a code with this level of complexity.

In this project, the aim is to explore a variety of cases of relevance to ITER and thus involving more complex physics to study the feasibility of the algorithm. Since the success of the parareal algorithm heavily relies on choosing the optimum coarse solver as a predictor, the project will involve studying various options for this purpose. The tasks will also include performing scaling studies to optimize the use of computing resources yielding maximum possible computational gain.