The dynamics of burning plasma is very complicated physics, which is dominated by multi-scale and multi-physics phenomena. To understand such phenomena, numerical simulations are indispensable. Fundamentals of numerical methods used in fusion science numerical modeling are briefly discussed in this paper. In addition, the parallelization technique such as open multi processing (OpenMP) and message passing interface (MPI) parallel programing are introduced and the loop-level parallelization is shown as an example.

Understanding the key processes occurring in the tokamak scrape-off layer (SOL) is becoming of the outmost importance while we enter the ITER era and we move towards the conception of future fusion reactors. By controlling the heat exhaust, by playing an important role in determining the overall plasma confinement, and by regulating the impurity level in tokamak core, the dynamics of the fusion fuel in the SOL is, in fact, related to some of the most crucial issues that the fusion program is facing today. Because of the limited diagnostic access and in view of predicting the SOL dynamics in future devices, simulations are becoming crucial to address the physics of this region. The present paper, which summarizes the lecture on SOL simulations that was given during the 7th ITER international school (August 25–29, 2014, Aix-en-Provence, France), provides a brief overview of the simulation approaches to the SOL dynamics. First, disentangling the complexity of the system, the key physics processes occurring in the SOL are described. Then, the different simulation approaches to the SOL dynamics are presented, from first-principles kinetic and fluid models, to the phenomenological analysis.

Gyrokinetic simulations of electromagnetic turbulence in magnetically confined torus plasmas including tokamak and heliotron/stellarator are reviewed. Numerical simulation of turbulence in finite beta plasmas is an important task for predicting the performance of fusion reactors and a great challenge in computational science due to multiple spatio-temporal scales related to electromagnetic ion and electron dynamics. The simulation becomes further challenging in non-axisymmetric plasmas. In finite beta plasmas, magnetic perturbation appears and influences some key mechanisms of turbulent transport, which include linear instability and zonal flow production. Linear analysis shows that the ion-temperature gradient (ITG) instability, which is essentially an electrostatic instability, is unstable at low beta and its growth rate is reduced by magnetic field line bending at finite beta. On the other hand, the kinetic ballooning mode (KBM), which is an electromagnetic instability, is destabilized at high beta. In addition, trapped electron modes (TEMs), electron temperature gradient (ETG) modes, and micro-tearing modes (MTMs) can be destabilized. These instabilities are classified into two categories: ballooning parity and tearing parity modes. These parities are mixed by nonlinear interactions, so that, for instance, the ITG mode excites tearing parity modes. In the nonlinear evolution, the zonal flow shear acts to regulate the ITG driven turbulence at low beta. On the other hand, at finite beta, interplay between the turbulence and zonal flows becomes complicated because the production of zonal flow is influenced by the finite beta effects. When the zonal flows are too weak, turbulence continues to grow beyond a physically relevant level of saturation in finite-beta tokamaks. Nonlinear mode coupling to stable modes can play a role in the saturation of finite beta ITG mode and KBM. Since there is a quadratic conserved quantity, evaluating nonlinear transfer of the conserved quantity from unstable modes to stable modes is useful for understanding the saturation mechanism of turbulence.

This paper explains how to obtain the distribution function of minority ions in tokamak plasmas using the Monte Carlo method. Since the emphasis is on energetic ions, the guiding-center transformation is outlined, including also the transformation of the collision operator. Even within the guiding-center formalism, the fast particle simulations can still be very CPU intensive and, therefore, we introduce the reader also to the world of high-performance computing. The paper is concluded with a few examples where the presented method has been applied.

The particle-in-cell (PIC) algorithm is the most popular method for the discretisation of the general 6D Vlasov–Maxwell problem and it is widely used also for the simulation of the 5D gyrokinetic equations. The method consists of coupling a particle-based algorithm for the Vlasov equation with a grid-based method for the computation of the self-consistent electromagnetic fields. In this review we derive a Monte Carlo PIC finite-element model starting from a gyrokinetic discrete Lagrangian. The variations of the Lagrangian are used to obtain the time-continuous equations of motion for the particles and the finite-element approximation of the field equations. The Noether theorem for the semi-discretised system implies a certain number of conservation properties for the final set of equations. Moreover, the PIC method can be interpreted as a probabilistic Monte Carlo like method, consisting of calculating integrals of the continuous distribution function using a finite set of discrete markers. The nonlinear interactions along with numerical errors introduce random effects after some time. Therefore, the same tools for error analysis and error reduction used in Monte Carlo numerical methods can be applied to PIC simulations.

Firstly, a brief overview will be given on different models that are able to describe the behaviour of wave propagation as a function of specific frequency ranges. Each range corresponds to different heating systems, namely, 20–100 MHz for the ion cyclotron resonant heating, 2–20 GHz for lower-hybrid heating or current drive, and 100–250 GHz for electron cyclotron resonant heating or current drive systems. The specification of every system will be explained in detail, including the typical set of equations and the assumptions needed to describe the properties of these heating or current drive systems, as well as their specific domains of validity. In these descriptions, special attention will be paid to the boundary conditions. A review of specific physical problems associated with the wave heating systems will also be provided. The review will detail the role of simulation in answering questions that arise from experiments on magnetized plasma devices devoted to fusion. A few examples that will be covered are the impact of edge turbulence on wave propagation and its consequences on heating system performance, the effects of fast particles and ponderomotive effects, among others. A study that is more focused on radio-frequency sheath effects will also be discussed. It shows that such simulations require very sophisticated tools to gain a partial understanding of the observations undertaken in dedicated experiments. To conclude this review, an overview will be given about the requirements and progress necessary to obtain relevant predictive simulation tools able to describe the wave heating systems used in fusion devices.

Immersed boundary methods for computing confined fluid and plasma flows in complex geometries are reviewed. The mathematical principle of the volume penalization technique is described and simple examples for imposing Dirichlet and Neumann boundary conditions in one dimension are given. Applications for fluid and plasma turbulence in two and three space dimensions illustrate the applicability and the efficiency of the method in computing flows in complex geometries, for example in toroidal geometries with asymmetric poloidal cross-sections.

Wavelet analysis and compression tools are reviewed and different applications for the study of MHD and plasma turbulence are presented. We introduce the continuous and the orthogonal wavelet transform and detail several statistical diagnostics based on the wavelet coefficients. We then show how to extract coherent structures out of fully developed turbulent flows using wavelet-based denoising. Finally some multiscale numerical simulation schemes using wavelets are described. Several examples for analysing, compressing and computing one-, two- and three-dimensional turbulent MHD or plasma flows are presented.

Challenges in plasma physics are wide. Investigation and advances are made in experiments but at the same time, to understand and to reach the experimental limits, accurate numerical simulations are required from systems of nonlinear equations. The numerical challenges of solving the associated fluid equations are discussed in this paper. Using the framework of the finite difference discretization, the most widely used methods for the problems linked to the diffusion or advection operators are presented.