Numerical modelling of strongly anisotropic dissipative effects in MHD

In magnetically confined fusion plasmas there is extreme anisotropy due to the high temperature and large magnetic field strength to the extent that thermal conductivity coefficients can be up to $10^{12}$ times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods. A common approach uses field aligned coordinates but in case of magnetic x-points and reconnection local non-alignment is unavoidable. Accuracy in case of high levels of anisotropy for non-field aligned grids is needed for the simulation of instabilities and radial transport processes in the presence of magnetic reconnection, e.g. with edge turbulence. % We therefore consider $2^{nd}$ order numerical schemes which are suitable for non-aligned grids. A novel method for co-located grids, developed to take into account the direction of the magnetic field, has been applied to the unsteady anisotropic heat diffusion equation on a non-field-aligned grid and compared with several other discretisation schemes, including G{\"{u}}nter et al's symmetric scheme. Test cases include variable diffusion coefficients with anisotropy values up to $10^{12}$, and field line bending in divergence and non-divergence free (unit vector) field configurations. % One of the model problems is given by the unsteady heat equation % \begin{equation*} \begin{split} \mbf{q} &= - D_\bot\nabla T - (D_\|-D_\bot)\mbf{b}\mbf{b}\cdot\nabla T, \quad \diff{T}{t} = -\nabla\cdot\mbf{q} + f, \end{split} \label{eq:braginskii} \end{equation*} where $T$ represents the temperature, \mbf{b} represents the unit direction vector of the magnetic field line with respect to the coordinate axes, $f$ is some source term and $D_\|$ and $D_\bot$ represent the parallel and the perpendicular diffusion coefficient respectively. \\ % Preliminary conclusions are that for FDM's the preservation of self-adjointness is crucial for limiting the pollution of perpendicular diffusion to acceptable values. However it is not required for maintaining the order of accuracy in most cases as is demonstrated by our aligned method. Key goal is to improve the co-located method to obtain acceptable levels for the pollution of the pe

Uncertainty quantification for random parabolic equations with non-homogeneous boundary conditions on a bounded domain via the approximation of the probability density function

[EN] This paper deals with the randomized heat equation defined on a general bounded interval [L-1, L-2] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L-1, L-2] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen-Loeve expansion, being Gaussian and non-Gaussian.This work has been supported by Spanish Ministerio de Economía y Competitividad grant MTM2017 89664 P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Uncertainty quantification for random parabolic equations with non-homogeneous boundary conditions on a bounded domain via the approximation of the probability density function. Mathematical Methods in the Applied Sciences. 42(17):5649-5667. https://doi.org/10.1002/mma.5333S564956674217Holden, H., Øksendal, B., Ubøe, J., & Zhang, T. (2010). Stochastic Partial Differential Equations. doi:10.1007/978-0-387-89488-1Casabán, M.-C., Company, R., Cortés, J.-C., & Jódar, L. (2014). Solving the random diffusion model in an infinite medium: A mean square approach. Applied Mathematical Modelling, 38(24), 5922-5933. doi:10.1016/j.apm.2014.04.063Xu, Z., Tipireddy, R., & Lin, G. (2016). Analytical approximation and numerical studies of one-dimensional elliptic equation with random coefficients. Applied Mathematical Modelling, 40(9-10), 5542-5559. doi:10.1016/j.apm.2015.12.041CalatayudJ CortésJC JornetM.On the approximation of the probability density function of the randomized heat equation.https://arxiv.org/pdf/1802.04190.pdfStrand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Vaart, A. W. van der. (1998). Asymptotic Statistics. doi:10.1017/cbo9780511802256Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Pitman, J. (1993). Probability. doi:10.1007/978-1-4612-4374-8Williams, D. (1991). Probability with Martingales. doi:10.1017/cbo9780511813658LawlessJF.Truncated Distributions: Wiley StatsRef: Statistics Reference Online;2014

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ 2012 Taylor & Francis.This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods

Nonlinear analysis of spacecraft thermal models

We study the differential equations of lumped-parameter models of spacecraft thermal control. Firstly, we consider a satellite model consisting of two isothermal parts (nodes): an outer part that absorbs heat from the environment as radiation of various types and radiates heat as a black-body, and an inner part that just dissipates heat at a constant rate. The resulting system of two nonlinear ordinary differential equations for the satellite's temperatures is analyzed with various methods, which prove that the temperatures approach a steady state if the heat input is constant, whereas they approach a limit cycle if it varies periodically. Secondly, we generalize those methods to study a many-node thermal model of a spacecraft: this model also has a stable steady state under constant heat inputs that becomes a limit cycle if the inputs vary periodically. Finally, we propose new numerical analyses of spacecraft thermal models based on our results, to complement the analyses normally carried out with commercial software packages.Comment: 29 pages, 4 figure

Modeling anisotropic diffusion using a departure from isotropy approach

There are a large number of finite volume solvers available for solution of isotropic diffusion equation. This article presents an approach of adapting these solvers to solve anisotropic diffusion equations. The formulation works by decomposing the diffusive flux into a component associated with isotropic diffusion and another component associated with departure from isotropic diffusion. This results in an isotropic diffusion equation with additional terms to account for the anisotropic effect. These additional terms are treated using a deferred correction approach and coupled via an iterative procedure. The presented approach is validated against various diffusion problems in anisotropic media with known analytical or numerical solutions. Although demonstrated for two-dimensional problems, extension of the present approach to three-dimensional problems is straight forward. Other than the finite volume method, this approach can be applied to any discretization method

Application of Meshless Methods for Thermal Analysis

Many numerical and analytical schemes exist for solving heat transfer problems. The meshless method is a particularly attractive method that is receiving attention in the engineering and scientific modeling communities. The meshless method is simple, accurate, and requires no polygonalisation. In this study, we focus on the application of meshless methods using radial basis functions (RBFs) — which are simple to implement — for thermal problems. Radial basis functions are the natural generalization of univariate polynomial splines to a multivariate setting that work for arbitrary geometry with high dimensions. RBF functions depend only on the distance from some center point. Using distance functions, RBFs can be easily implemented to model heat transfer in arbitrary dimension or symmetry