In magnetized plasma, the magnetic field confines the particles around the field lines. The anisotropy intensity in the viscosity and heat conduction may reach the order of $10^{12}$. When the boundary conditions are periodic or Neumann, the strong diffusion leads to an ill-posed limiting problem. To remove the ill-conditionedness in the highly anisotropic diffusion equations, we introduce a simple but very efficient asymptotic preserving reformulation in this paper. The key idea is that, instead of discretizing the Neumann boundary conditions locally, we replace one of the Neumann boundary condition by the integration of the original problem along the field line, the singular $1/ε$ terms can be replaced by $O(1)$ terms after the integration, so that yields a well-posed problem. Small modifications to the original code are required and no change of coordinates nor mesh adaptation are needed.
Uniform convergence with respect to the anisotropy strength $1/ε$ can be observed numerically and the condition number does not scale with the anisotropy.
In this paper, a new asymptotic preserving (AP) scheme is proposed for the anisotropic elliptic equations. Different from previous AP schemes, the actual one is based on first-order system least-squares for second-order partial differential equations, and it is uniformly well-posed with respect to anisotropic strength. The numerical computation is realized by a deep neural network (DNN), where least-squares functionals are employed as loss functions to determine parameters of DNN. Numerical results show that the current AP scheme is easy for implementation and is robust to approximate solution…
Efficient and robust iterative solvers for strong anisotropic elliptic equations are very challenging. In this paper a block preconditioning method is introduced to solve the linear algebraic systems of a class of micro-macro asymptotic-preserving (MMAP) scheme. MMAP method was developed by Degond {\it et al.} in 2012 where the discrete matrix has a $2\times2$ block structure. By the approximate Schur complement a series of block preconditioners are constructed. We first analyze a natural approximate Schur complement that is the coefficient matrix of the original non-AP discretization. However…
We propose and analyze structure-preserving parametric finite element methods (SP-PFEM) for evolution of a closed curve under different geometric flows with arbitrary anisotropic surface energy $γ(\boldsymbol{n})$ for $\boldsymbol{n}\in \mathbb{S}^1$ representing the outward unit normal vector. By introducing a novel surface energy matrix $\boldsymbol{G}_k(\boldsymbol{n})$ depending on $γ(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbolξ$-vector as well as a nonnegative stabilizing function $k(\boldsymbol{n}):\ \mathbb{S}^1\to \mathbb{R}$, which is a sum of a symmetric positive definite matr…
We construct a new nonlinear finite volume (FV) scheme for highly anisotropic diffusion equations, that satisfies the discrete minimum-maximum principle. The construction relies on the linearized scheme satisfying less restrictive monotonicity conditions than those of an M-matrix, based on a weakly regular matrix splitting and using the Cartesian structure of the mesh (extension to quadrilateral meshes is also possible). The resulting scheme, obtained by expressing fluxes as nonlinear combinations of linear fluxes, has a larger stencil than other nonlinear positivity preserving or minimum-maxi…
An asymptotic preserving scheme for strongly anisotropic elliptic problems
Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse（图卢兹大学，UPS，INSA，UT1，UTM，图卢兹数学研究所）；CNRS, Institut de Mathématiques de Toulouse UMR 5219（法国国家科学研究中心，图卢兹数学研究所 UMR 5219）；CMI/LATP, Université de Provence（数学中心/分析与应用数学实验室，普罗旺斯大学）
Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations
Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse（图卢兹大学，UPS，INSA，UT1，UTM，图卢兹数学研究所）；CNRS, Institut de Mathématiques de Toulouse UMR 5219（法国国家科学研究中心，图卢兹数学研究所 UMR 5219）；CMI/LATP, Université de Provence（CMI/LATP，普罗旺斯大学）
An Asymptotic-Preserving method for highly anisotropic elliptic equations based on a micro-macro decomposition
Université de Toulouse, UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse（图卢兹大学，UPSMATHS）；CNRS, Institut de Mathématiques de Toulouse UMR 5219（法国国家科学研究中心，图卢兹数学研究所UMR 5219）；CMI/LATP, Université de Provence（数学与理论物理中心/分析、应用与概率实验室，普罗旺斯大学）
A Structure-Preserving Symmetric Galerkin Boundary Element Method for Anisotropic Multi-phase Mullins-Sekerka Flow
Université Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208（里昂第一大学、法国国家科学研究中心、中央里昂学院、里昂国立应用科学学院、让·莫内大学、ICJ UMR5208）
The anisotropic and heterogeneous $N$-dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. A recent structure-preserving mixed Galerkin method is applied, leading directly to a finite-dimensional port-Hamiltonian system: its numerical analysis is carried out in a general framework. Optimal choices of mixed finite elements are then proved to reach the best trade-off between the convergence rate and the number of degrees of freedom for the state error. Exta compatibility conditions are identified for the Hamiltonian error to be twice tha…
How to effectively remove the noise while preserving the image structure features is a challenging issue in the field of image denoising. In recent years, fractional PDE based methods have attracted more and more research efforts due to the ability to balance the noise removal and the preservation of image edges and textures. Among the existing fractional PDE algorithms, there are only a few using spatial fractional order derivatives, and all the fractional derivatives involved are one-sided derivatives. In this paper, an efficient feature-preserving fractional PDE algorithm is proposed for im…
Modified Patankar (MP) schemes are conservative, linear implicit and unconditionally positivity preserving time-integration schemes constructed for production-destruction systems. For such schemes, a classical stability analysis does not yield any information about the performance. Recently, two different techniques have been proposed to investigate the properties of MP schemes. In Izgin et al. [ESAIM: M2AN, 56 (2022)], inspired from dynamical systems, the Lyapunov stability properties of such schemes have been investigated, while in Torlo et al. [Appl. Numer. Math., 182 (2022)] their oscillat…
In this work, we present a modification of explicit Runge-Kutta temporal integration schemes that guarantees the preservation of any locally-defined quasiconvex set of bounds for the solution. These schemes operate on the basis of a bijective mapping between an admissible set of solutions and the real domain to strictly enforce bounds. Within this framework, we show that it is possible to recover a wide range of methods independently of the spatial discretization, including positivity preserving, discrete maximum principle satisfying, entropy dissipative, and invariant domain preserving scheme…
Higher-order time integration methods that unconditionally preserve the positivity and linear invariants of the underlying differential equation system cannot belong to the class of general linear methods. This poses a major challenge for the stability analysis of such methods since the new iterate depends nonlinearly on the current iterate. Moreover, for linear systems, the existence of linear invariants is always associated with zero eigenvalues, so that steady states of the continuous problem become non-hyperbolic fixed points of the numerical time integration scheme. Altogether, the stabil…
Splitting schemes are numerical integrators for Hamiltonian problems that may advantageously replace the Störmer-Verlet method within Hamiltonian Monte Carlo (HMC) methodology. However, HMC performance is very sensitive to the step size parameter; in this paper we propose a new method in the one-parameter family of second-order of splitting procedures that uses a well-fitting parameter that nullifies the expectation of the energy error for univariate and multivariate Gaussian distributions, taken as a problem-guide for more realistic situations; we also provide a new algorithm that through an …
In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for chemical reactive flows. There are several difficulties in constructing suitable numerical schemes. First of all, the density and internal energy are positive, and the mass fraction of each species is between 0 and 1. Secondly, due to the rapid reaction rate, the system may contain stiff sources, and the strong-stability-preserving explicit Runge-Kutta method may result in limited time step sizes. To obtain physically relevant numerical approximations, we apply the bound-preserving technique to the DG methods. F…
Embedded Trefftz DG method for reaction-diffusion problems on anisotropic meshes
University of Milano–Bicocca（米兰比可卡大学）；IMATI-CNR “Enrico Magenes”（IMATI-CNR“恩里科·马盖斯”研究所）；University of Pavia（帕维亚大学）；University of Vienna（维也纳大学）；Georg-August-Universität Göttingen（哥廷根乔治-奥古斯特大学）
For the evolution of a closed surface under anisotropic surface diffusion with a general anisotropic surface energy $γ(\boldsymbol{n})$ in three dimensions (3D), where $\boldsymbol{n}$ is the unit outward normal vector, by introducing a novel symmetric positive definite surface energy matrix $\boldsymbol{Z}_k(\boldsymbol{n})$ depending on a stabilizing function $k(\boldsymbol{n})$ and the Cahn-Hoffman $\boldsymbolξ$-vector, we present a new symmetrized variational formulation for anisotropic surface diffusion with weakly or strongly anisotropic surface energy, which preserves two important str…
