Transient phenomena and their control are of high relevance in magnetic confinement fusion plasmas to guarantee a stable and safe plasma operation. Interpretative simulations can maximize the insights gained from experiments on present machines and predictive simulations can help in the preparation of design, mitigation techniques and operational scenarios for future devices. In this article, we provide an overview of recent advances and novel scientific results obtained with the 3D non-linear hybrid fluid-kinetic code JOREK, covering physics of plasma transients from the core to the scrape-off layer (SOL) both for tokamak and stellarator devices. Substantial progress was made in the physics understanding, model validation with experiments and experiment interpretation, thus, giving confidence for predictions to devices like DTT, ITER and DEMO. The topics addressed comprise a wide range: the edge physics of new operation scenarios and edge localized mode suppression; major disruptions with a focus on runaway electrons and vertical displacement events as well as disruption mitigation by shattered pellet injection; the physics mechanisms and operational limits of the flux pumping regime for sawtooth control; MHD limits of stellarators and work towards incorporating advanced edge/SOL/exhaust dynamics; continuing improvements of the code for more efficient hybrid simulations on conventional and accelerated high performance computing architectures.

We present a matrix-free parallel scalable multilevel deflation preconditioned method for heterogeneous time-harmonic wave problems. Building on the higher-order deflation preconditioning proposed by Dwarka and Vuik (SIAM J. Sci. Comput. 42(2):A901-A928, 2020; J. Comput. Phys. 469:111327, 2022) for highly indefinite time-harmonic waves, we adapt these techniques for parallel implementation in the context of solving large-scale heterogeneous problems with minimal pollution error. Our proposed method integrates the Complex Shifted Laplacian preconditioner with deflation approaches. We employ higher-order deflation vectors and re-discretization schemes derived from the Galerkin coarsening approach for a matrix-free parallel implementation. We suggest a robust and efficient configuration of the matrix-free multilevel deflation method, which yields a close to wavenumber-independent convergence and good time efficiency. Numerical experiments demonstrate the effectiveness of our approach for increasingly complex model problems. The matrix-free implementation of the preconditioned Krylov subspace methods reduces memory consumption, and the parallel framework exhibits satisfactory parallel performance and weak parallel scalability. This work represents a significant step towards developing efficient, scalable, and parallel multilevel deflation preconditioning methods for large-scale real-world applications in wave propagation.

We propose a matrix-free parallel two-level deflation method combined with the Complex Shifted Laplacian Preconditioner (CSLP) for two-dimensional heterogeneous Helmholtz problems encountered in seismic exploration, antennas, and medical imaging. These problems pose challenges in terms of accuracy and convergence due to scalability issues with numerical solvers. Motivated by the limitations imposed by excessive computational time and memory constraints when employing a sequential solver with constructed matrices, we parallelize the two-level deflation method without constructing any matrices. Our approach utilizes preconditioned Krylov subspace methods and approximates the CSLP preconditioner with a parallel geometric multigrid V-cycle. For the two-level deflation, standard inter-grid deflation vectors and further high-order deflation vectors are considered. As another main contribution, the matrix-free Galerkin coarsening approach and a novel re-discretization scheme as well as high-order finite-difference schemes on the coarse grid are studied to obtain wavenumber-independent convergence. The optimal settings for an efficient coarse-grid problem solver are investigated. Numerical experiments of model problems show that the wavenumber independence has been obtained for medium wavenumbers. The matrix-free parallel framework shows satisfactory weak and strong parallel scalability.

In recent years, domain decomposition based preconditioners have become popular tools to solve the Helmholtz equation. Notorious for causing a variety of convergence issues, the Helmholtz equation remains a challenging PDE to solve numerically. Even for simple model problems, the resulting linear system after discretisation becomes indefinite and tailored iterative solvers are required to obtain the numerical solution efficiently. At the same time, the mesh must be kept fine enough in order to prevent numerical dispersion ‘polluting’ the solution [4]. This leads to very large linear systems, further amplifying the need to develop economical solver methodologies.

Finding fast yet accurate numerical solutions to the Helmholtz equation remains a challenging task. The pollution error (i.e. the discrepancy between the numerical and analytical wave number k) requires the mesh resolution to be kept fine enough to obtain accurate solutions. A recent study showed that the use of Isogeometric Analysis (IgA) for the spatial discretization significantly reduces the pollution error. However, solving the resulting linear systems by means of a direct solver remains computationally expensive when large wave numbers or multiple dimensions are considered. An alternative lies in the use of (preconditioned) Krylov subspace methods. Recently, the use of the exact Complex Shifted Laplacian Preconditioner (CSLP) with a small complex shift has shown to lead to wave number independent convergence while obtaining more accurate numerical solutions using IgA. In this paper, we propose the use of deflation techniques combined with an approximated inverse of the CSLP using a geometric multigrid method. Numerical results obtained for one- and two-dimensional model problems, including constant and non-constant wave numbers, show scalable convergence with respect to the wave number and approximation order p of the spatial discretization. Furthermore, when kh is kept constant, the proposed approach leads to a significant reduction of the computational time compared to the use of the multigrid-approximated or exact inverse of the CSLP with a small shift, in particular for three-dimensional model problems.

Recent research efforts aimed at iteratively solving the Helmholtz equation have focused on incorporating deation techniques for accelerating the convergence of Krylov subpsace methods. The requisite for these efforts lies in the fact that the widely used and well-acknowledged complex shifted Laplacian preconditioner (CSLP) shifts the eigenvalues of the preconditioned system towards the origin as the wave number increases. The two-level-deation preconditioner combined with CSLP showed encouraging results in moderating the rate at which the eigenvalues approach the origin. However, for large wave numbers the initial problem resurfaces and the near-zero eigenvalues reappear. Our findings reveal that the reappearance of these near-zero eigenvalues occurs if the near-singular eigenmodes of the fine-grid operator and the coarse-grid operator are not properly aligned. This misalignment is caused by accumulating approximation errors during the inter-grid transfer operations. We propose the use of higher-order approximation schemes to construct the deation vectors. The results from rigorous Fourier analysis and numerical experiments confirm that our newly proposed scheme outperforms any other deation-based preconditioner for the Helmholtz problem. In particular, the spectrum of the adjusted preconditioned operator stays fixed near one. These results can be generalized to general shifted indefinite systems with random right-hand sides. For the first time, the convergence properties for very large wave numbers (k = 106 in one dimension and k = 103 in two dimensions) have been studied, and the convergence is close to wave number independence. Wave number independence for three dimensions has been obtained for wave numbers up to k = 75. The new scheme additionally shows very promising results for the more challenging Marmousi problem. Irrespective of the strongly varying wave number, we obtain a constant number of iterations and a reduction in computational time as the results remain robust without the use of the CSLP preconditioner.

Recent research efforts aimed at iteratively solving the Helmholtz equation have focused on incorporating deflation techniques for accelerating the convergence of Krylov subspace methods. In this work, we extend the two-level deflation method in [6] to a multilevel deflation method. By using higher-order deflation vectors, we show that up to the level where the coarse-grid linear systems remain indefinite, the near-zero eigenvalues of the these coarse-grid operators remain aligned with the fine-grid operator keeping the spectrum of the preconditioned system fixed away from the origin. Combining this with the well-known CSLP-preconditioner, we obtain a scalable solver with theoretical linear complexity for the highly indefinite Helmholtz equation. This can be attributed to a fixed number of iterations independent of the wave number and an optimal use of the CSLP-preconditioner. We approximate the CSLP-preconditioner, while allowing the complex shift to be small. The proposed configuration additionally shows very promising results for the more challenging Marmousi problem.