## 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.

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).