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 .


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.
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?
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).