Quasilinear Heat Equation in Three Dimensions and Stefan Problem in Permafrost Soils in the Frame of Alternating Directions Finite Difference Scheme

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Abstract — The quasilinear heat equation with thermal conductivity and heat capacity depending on the temperature field in three spatial dimensions is studied in application to the phase transition problem in permafrost soils. The conditions under which the alternating directions Douglas – Rachford finite difference scheme retains numerical stability are explicitly formulated. The comparison with the known analytical similarity solution to the Stefan problem in one spatial dimension is performed.

Index Terms—quasilinear heat equation, Stefan problem, finite differences, alternating directions scheme, numerical stability.


Since the early formulations of the alternating directions implicit methods (ADI methods) [1], [2], they have been tremendously developed and found a vast number of applications [3], [4]. Nevertheless, serious difficulties are encountered with the use of these methods in application to problems with complex geometries [5] and/or nonlinear equations of mathematical physics [6].

Although the schemes of the ADI methods are proved to be efficient and economic with respect to time consumption and, in most cases, unconditionally stable, they exhibit some disadvantages:

1) Their finite-difference formulations permit to consider only rectangular spatial domains (due to commutativity requirements imposed on the factorized and split operators) [7];

2) The application of the ADI schemes to the problems with Neumann and Robin boundary conditions that are varying in time encounters serious problems due to the necessity of evaluation or approximation of these boundary conditions at the intermediate steps of the scheme [8];

3) When applied to the solution of nonlinear heat equations, the operators constituting an ADI scheme do not commute, thus leading to the loss of unconditional stability of the scheme [6].

The first of the above disadvantages can be overcome either by the use of finite elements methods in conjunction with operator splitting techniques, or by domain decomposition techniques. The second and the third disadvantages present an important problem for the successful application of the ADI scheme. To the best of our knowledge, no complete stability analysis for an ADI scheme applied to the nonlinear heat equation in a three-dimensional spatial domain is available in the literature, thus motivating this work.

Another motivation for the present work is the application of ADI scheme to the modeling of heat transfer in large scale environmental systems (e.g., large areas of permafrost soils) which, in the case of purely explicit finite-differences schemes, imposes stiff constraints on the time-step in order to guarantee the numerical stability. At the same time, implementation of implicit schemes can often lead to much greater computing expenses than that of explicit schemes, especially for the problems with rapidly changing coefficients in complex geometries and substantially nonhomogeneous meshes. Thus, in modeling of heat transfer in large scale systems the necessity of making an optimal choice between explicit and implicit schemes arises. In case of finite-elements method, applied to modeling of processes in permafrost soils, the analysis of numerical stability appears to be so complex, that the stability criterion is often established empirically [9].

In the present paper we discuss the application of the ADI Douglas – Rachford scheme to the solution of Stefan problem in porous permafrost soils. The paper is organized as follows: next section contains the problem formulation and some assumptions that will be used in the proof of numerical stability of the ADI scheme while section 3 exposes the proof itself. Section 4 presents some numerical results and is followed by Conclusions.

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Alternate directions implicit scheme and the intermediate boundary conditions. Third type boundary conditions.

This second Note is dedicated to the discussion of the case of 3-rd type boundary conditions posed on the edges of the modeling region and will present an outline of the algorithm for the account of both the Dirichlet and 3-rd type boundary conditions on different edges.

Before passing to the discussion of that scheme, a few words should be said about another ADI scheme – namely 2D Peaceman – Rachford finite difference scheme. The correct account of the intermediate boundary conditions for both Dirichlet and 3-rd order types has been thoroughly discussed in [1]. One essential difference between the 3D Douglas – Rachford and 2D Peaceman – Rachford schemes is that in the latter scheme the spatial coordinates enter symmetrically, such that the second equation (in 2D Peaceman – Rachford) contains finite difference representations of both spatial derivatives. This fact leads to a cumbersome infer of relation between the values of the unknown function on different edges – for details see [1].

Surprisingly, although the Douglas – Rachford scheme is designed for a 3D spatial region, it is easier to treat the 3-rd order boundary conditions for it correctly due to the “non-symmetrical” entrance of the spatial coordinates in three equations – see Eq. (1) – (3):

, (1)
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Alternate directions implicit scheme and the intermediate boundary conditions

The ADI (alternate directions implicit) method is widely used for the numerical solution of multidimensional parabolic PDE (partial differential equations). [1]. Although the method is known for a long time and is well described in the text-books, its practical realizations sometimes appear to be inaccurate [2]. The inaccuracy arises every time when one neglects the correct account of the so-called intermediate boundary conditions. This neglect can become the cause of instabilities even when the used ADI-scheme is known to be unconditionally stable in the frame of von Neumann spectral analysis technique [3]. The procedures of a correct account of the intermediate boundary conditions (for the Peaceman-Rachford ADI-scheme), are described in [4, 5, 6].
Below, we are going to consider the correct account of the intermediate boundary conditions for the Douglas-Rachford ADI-scheme [3, 7]:

, (1) 
, (2) 
, (3) 
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