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 , , , , , , , .

With we have
, . In order to find при , let us use implicit difference scheme (about explicit scheme see remark 2 listed below)
,
where .It is a generalization of the introduced scheme 1 for the coefficients equation not dependent on x. Suppose that partial derivatives , re bounded and functions

(4)

are distributed at the whole field by Taylor formula with remainder terms , and respectively. Using terms 2 and carrying out arguments similar to the arguments listed below, we can prove approximation scheme error .

To approximate the heat flow in the left boundary condition (3) right difference derivative can be used
. (5)

Meanwhile approximation error would have had just first sequence in space.

Universal way to improve this procedure is the introduction of phantom (ghost, imaginary, fictitious) node [2, 3] out of modeling area (see Fig. 1) and the use of central difference derivative .

 

Fig. 1. Phantom node x_1

It is also possible to introduce approach based on the account of heat equation for the boundary node. In [4], this approach is applied to the linear heat equation with constant coefficients. Below it is generalized in case of nonlinear heat equation.

Fix the moment of time , . Rewrite condition (3), using the last notations (4):

. (6)

Build difference approximation with error . From Taylor expansion при we have
. (7)

Let us derivate difference approximation and for their further substitution in (7).

By definition
. (8)

By Taylor formula , whence .It is equal to the fact, that approximates factor into (8) with error . At the same time for a second factor it is done . Using limitation previously assumed and limitation K (T),specified in clause (2), from the equality we have

  (9)

Only approximation is left. By definition . Using heat conduction equation (1), replace the right side of this equation:

. (10)

By analogy with derivation from (8) equality (9), we obtain from (10) the equality

. (11)

Substitute approximations (9) and (11) into equality (7), and the result of this substitution – into boundary condition (6). Neglecting terms of order , finally we have

. (12)

At this coincides with [4].

Rewrite condition (12) as


(13)

Remark 1. From (13) it is obvious, that coefficient at exceeds the absolute value at .It ensures strict diagonal dominance [5] of difference equations matrix system.

Remark 2. Approximation (13) gives the best fit to using with an explicit difference scheme. It is also possible by analogy with [4] use the ratio
. (14)

The approximation error in this case will also be .

References
1. A. А. Samarsky. Difference schemes theory . – Мoscow: Nauka, 1977.
2. T. J. Chung. Computational fluid dynamics. – Cambridge: Cambridge University Press, 2002.
3. N. N. Kalitkin. Numerical methods. – Мoscow: Nauka, 1978.
4. A. А. Samarsky. Introduction to difference schemes theory. – Мoscow: Nauka, 1977.
5. R.A. Horn, Charles R. Johnson. Matrix analysis. – Мoscow:Mir, 1989.

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

  1. Friendly remark,
    The formulae do look very readbable and are non-standard.

    I miss how the paper linearises the nonlinear PDE and the issue of (conditional) stability. Do you have numerical experiments for this problem?

  2. Dear Daniel, these non-standard-looking formulae are obtained via the approach, described in "The Theory of Difference Schemes" by Alexander A. Samarski, pp. 82 -- 84.
    The equation is linearized in the usual way (for explicit finite-difference schemes): one takes the values of temperature-dependent coefficients from the previous time level. The stability issue is managed using sufficiently small time step (see method of frozen coefficients).
    As for numerical experiments, we have carried out a lot. The results show that for thin space grids the described boundary condition approximation (of second order in space) is much more effective than the simple one (of first order).

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