A new tailored finite point method for strongly anisotropic diffusion equation on misaligned grids
Introduction
Anisotropic diffusion equations has been arisen in a wide range of scientific fields, including the radiation hydrodynamics [1], [2], [3], heat conduction in fusion plasmas [4], [5], [6], flows in porous media [7], atmospheric or oceanic flows [8] and so on. However, the numerical resolution of highly anisotropic physical problems is still a challenging task. It is not always possible or desirable to align the computational grid with the principal axes of transport. However, when the grid is not aligned, it becomes difficult to represent the model accurately, and a numerical cross-flux may be generated that pollutes the solution. As summarized in [6], several problems may arise: (1) yield anisotropy dependent convergence; (2) introduce large perpendicular errors; (3) loss positivity near high gradients.
Some scientists studied the efficient numerical methods on anisotropic diffusivity [9]. Most of the techniques to handle anisotropic diffusion equations are based on finite volume method [2], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], finite difference methods [4], [5], [6], mimetic finite difference methods [21], finite element methods [22], [23], [24], Discontinuous Galerkin methods [25], [26] and so on. The development of the numerical methods of these problems deal with the important properties range from the classical stability and accuracy to some other desirable numerical properties, including simplicity, robustness, cost-efficiency, local stencil, local conservation, positivity preserving or monotonicity etc.
Finite volume schemes are methods of choice for a number of engineering applications. It allows us to obtain the local conservativity of the fluxes which is significant in physics. One of the most popular method is the Multi Point Flux Approximation (MPFA) [10], [11], [12], [13], [14], [15]. Let us mention Aavatsmark [10], [11] presents a symmetric MPFA method for quadrilateral grids in two and three dimensions and give formal proof of convergence. Edwards and Pal [12], [15] develop a family control volume distributed multi-point flux approximation (CVD-MPFA) schemes for solving the anisotropic diffusion equation on structured and unstructured grids. And the method is robust in terms of diffusion tensor discontinuity. Edwards and Zheng [13], [14] use M-matrix conditions and temperature continuity and flux continuity conditions to obtain a suitable quadrature point for the flux formulation.
Besides, positivity preserving or monotonicity is one of the key requirements to avoid non-physical negative temperature for diffusion equation. There are many literatures devoted to monotone schemes [18], [19], [20], [27], [28]. Algorithms based on slope limiters are proposed to avoid negative temperature in the presence of large temperature gradients in [27]. The gradient is calculated by using a nonlinear way to construct a finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes in [28]. Sheng and Yuan [18], [19] introduce nonnegative parameters to define nonlinear flux to obtain nonlinear finite volume scheme preserving positivity for diffusion equations.
There is another difficulty which relates to the boundary conditions are periodic or Neumann. The strong diffusion leads to an ill-posedness system when the anisotropy goes to infinity. This system has been studied in a series of papers by Degond et al. [29], [30], [31]. Their methods are referred to as asymptotic preserving (AP) method that works for all cases that the strong of the anisotropy δ ranges from O(1) to 1020. In particular, Castillo and Chacn [32] used field line integration to help solve anisotropic diffusion problems of the extreme anisotropic diffusion. The Lagrangian Greens function method has been proposed for the accurate and efficient computation of the strongly anisotropic diffusion equation. Based on that, L Chacn et al. developed an AP scheme that can deal with all ranges of δ based on the Lagrangian Greens function method [33].
The purpose of this study is to develop a new TFPM to deal with the strongly anisotropic diffusivity. TFPM was first introduced by Han, Huang and Kellogg for singular perturbed elliptic equations with constant coefficients [34], [35], [36]. The essential idea of the TFPM is to locally approximate the coefficients of the equation by piecewise constants functions firstly. Then use the exact solutions of the constant coefficient equation as local basis functions to formulate a discrete linear system. The TFPM builds in the properties of the solution, this modification not only maintain essential features but also obtain more accurate approximation to the original problem. TFPM is able to achieve higher accuracy than standard methods but at a reasonable cost to capture rapid transitions in a numerical approximation without using a large number of unknowns.
The problem under consideration is the following two dimensional steady-state diffusion reaction equation with anisotropic diffusivity:Here Ω ⊂ R2 is a domain, with boundary ∂Ω; f(x) ∈ L2(Ω), the diffusion tensor A is a symmetric positive definite matrix given bywith θ may vary with space, α1, α2 are constant. Let . The smaller ϵ is, the stronger the anisotropy.
A novel methodology for the solution of the anisotropic diffusion problem is presented in this paper. Compared with the TFPM in previous work [34], the key point in the construction of our scheme is the discretization of the flux across each cell edge. Let Γ is the common edge of two adjacent cells. The following conditions must be satisfied at the interface between the two subdomains: and where n is the normal direction of Γ.
The arrangement of this paper is as follows. In Section 2, we construct a new TFPM based on the interface condition to deal with strongly anisotropic diffusivity. Three numerical examples are presented in Section 3. Uniform convergence can be observed for the solution. And the new TFPM can sharply capture internal layers. Finally we conclude this paper in Section 4.
Section snippets
The TFPM for the misalignment angle θ is constant
Let . Eq. (1) can be written in the coordinates (ξ, η) aligned with the transport axes asWithout loss of generality we assume that α1 > α2, so that the strong transport direction is aligned with ξ. Considering Eq. (1) in Cartesian coordinates (x, y) that are not aligned with the transport axes, and where the misalignment angle between x and ξ is θ. Let . Then Eq. (1) can be
Numerical experiments
In the subsequent part, we present three computational experiments to demonstrate the performance of the new TFPM. For all the examples, . The computational domain is .
Example one: As an initial test we consider a simple steady diffusion problem. The imposed exact solution reads: where the angle of misalignment θ is set to be a constant. Computational results for this test case are given in Fig. 2. It is demonstrated that the new scheme conserve the
Conclusion
We introduce a new TFPM for strongly anisotropic tensor diffusivity in this paper. Numerically, uniform second order convergence can be observed. Compared with SM9, the new TFPM shows the superior of internal problem. Some future works include the application to practical multi-material physical problems, the extension to unstructured triangular and polygonal meshes to meet the complicated computation domain.
Acknowledgments
This work was partially supported by Xu Lun Scholars Programme, Shanghai Lixin University of Accounting and Finance. Science Challenge Project (No. TZ2016002) and the National Natural Science Foundation of China (No. 11601328). The authors would like to thank the anonymous referees for their useful suggestions.
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