Application of Static Adaptive Partitioning of the Computational Domain

Introduction

Many software packages for numerical computations allow users to use a static adaptive (hereby referred to as adaptive) step in the construction of an orthogonal hexahedral structured computational mesh. This means that informed users can employ their experience to get a more accurate computation without significantly increasing the computation time by specifying the areas of the computational domain in which, in their opinion, it is necessary to apply more detailed partitioning (use a smaller spatial step) as compared to the rest of the computational domain.

When properly used, adaptive partitioning of the computational domain is a powerful tool in numerical computations to increase accuracy. However, when the above option is overused, the computational time can increase dramatically without necessarily altering the accuracy of the computation to any significant degree. In this article, we describe the theoretical advantages and disadvantages of using adaptive partitioning of the computational domain, and also give two examples for numerical computations of thermal fields in soils. In the first example, application of the adaptive step is appropriate; this is not the case, however, in the second.

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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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Finite-difference approximation of the boundary conditions of the second and third order for the nonlinear heat conduction equation

Let us consider non-linear heat conduction equation

, (1)

where
, (2)

with the initial condition , and boundary conditions
, (3)

At . If , condition (3) is the boundary condition of the second order, but if , it is the boundary condition of the third order.

Let us introduce analytical grids in space and time:, , , .

Then, define grid functions , , , , , , , .

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