In the field of the numerical Analysis, one can have to seek the solution of a partial derivative equation. Among the methods of usually practiced resolutions, the method the finite differences is easiest of access, since it rests on two concepts: discretization of the operators of derivation/differentiation (enough intuitive) on the one hand, and the convergence of the numerical diagram thus obtained on the other hand.

Approximation of the operators by formulas of Taylor

Thanks to the formulas of Taylor, one defines the discretization of the differential operators (derivative first, seconds, etc, partial or not).

The formulation of Taylor-Young is preferable in his simple use, the formulation of Taylor with integral remainder of Laplace makes it possible to measure the errors (lower cf)

Example of approximation of operator

For example, let us write in a point $x$ and for a value $h$ such as $u$ three times are derivable on an interval containing strictly $x+h$, one can write the two formulas of Taylor-Young:

$u\; (X\; +\; H)\; =\; U\; (X)\; +\; \backslash \; sum\_\; \{N\; =\; 1\}\; ^3\; \{\backslash \; frac\; \{h^n\}\; \{N!\; \}\; u^\; \{(N)\}(X)\}\; +\; h^3\; \backslash \; varepsilon\_1\; (X,\; H)$

and:

$u\; (X\; -\; H)\; =\; U\; (X)\; +\; \backslash \; sum\_\; \{N\; =\; 1\}\; ^3\; \{\backslash \; frac\; \{(-\; H)\; ^n\}\; \{N!\; \}\; u^\; \{(N)\}(X)\}\; +\; h^3\; \backslash \; varepsilon\_2\; (X,\; H)$

where all the applications $\backslash \; varepsilon\_i$ converge towards 0 with $h$. Then:

$\backslash \; frac\; \{U\; (X\; +\; H)\; -\; U\; (X)\}\; \{H\}\; =\; u\text{'}\; (X)\; +\; \backslash \; frac\; \{H\}\; \{2\}\; U$ (X) + \ frac {h^2} {3!} u^ {(3)}(X) + h^2 \ varepsilon_1 (X, H)

and by summoning the developments for x-h and x+h one obtains:

$\backslash \; frac\; \{U\; (X\; +\; H)\; -\; U\; (X\; -\; H)\}\; \{2:00\}\; =\; u\text{'}\; (X)\; +\; \backslash \; frac\; \{h^2\}\; \{6\}\; u^\; \{(3)\}(X)\; +\; h^2\; \backslash \; varepsilon\_3\; (X,\; H)$

one respectively obtains approximations of 1st order and 2nd order in $h$.

Grid

A grid is a whole of points of the field of definition to which one will apply the method the finite differences. For an application defined on a segment of $\backslash \; mathbb\; R$, one will in general add the two ends of the segment; for a grid in higher dimension, one will be brought to choose, possibly, of the points of contours of the field of definition.

One calls the step of the grid the distance between two successive points of the grid neighbors. In dimension 1, that is simplified of difference of the X-coordinates. This step is not necessarily constant, it can even be judicious not to fix it as tel. the step (total) of the approximation can be defined like the greatest step of the grid. Thus, if this total step tends towards 0, that wants to say that the distribution of the points of the grid in the selected interval tends to be done on all the field of study by density.

Example of grid

For an interval of validity $0,1$ one will use $(M+1)$ points, for example $\backslash \; \{0,\; H,\; 2:00,\dots ,\; M\; H\; =\; 1\; \backslash \}$ for a constant step $h\; =\; \backslash \; frac\; \{1\}\; \{M\}$.

Degree of derivation

For reasons at the same time of algebraic writing and study of convergence/stability a priori it is important to be replaced as much as possible in problems with orders of derivation weakest possible, even if it means to increase the dimension of the space of study. One will call upon intermediate variables thus: the derivative or derivative partial of the initially studied functions.

Example of lowering of degree of derivation

$u$ (X) - \ omega^2 U (X) = 0, U (0) = u_0, u' (0) = u_1

will be written readily with the choice:

$U\; (X)\; =\; (U\; (X),\; u\text{'}\; (X))$

what will give the equation for $U$:

$U\text{'}\; (X)\; +\; HAS\; U\; (X)\; =\; 0,\; U\; (0)\; =\; (u\_0,\; u\_1)$

where the matrix has is:

Thus the degree of derivation is tiny room to 1 here, whereas the dimension of the space of arrival became 2.

Numerical diagram

To write a numerical diagram of resolution of the initial differential equation means:

• subsituer formulations of derived/differential obtained by approximation with the operators themselves on all the points of the grid.

• to reorganize the equations to reveal an explicit diagram (ex: values at the date t+1 given according to the values of dates 0 to T) or implicit (an equation binds the values passed, present and future without one managing to express these last alone).

Within a traditional framework of modeling linear operators in linear differential equations, one leads to a system of linear equations of size equal to the number of nodes of the grid (in fact a little less, because of the initial data, for example).

To solve the numerical diagram simply means to find the values discrete of the function in each nodes.

a system resulting from a linear equation can often be algebraically simple to solve. To simplify, one can say that the explicit diagrams generate systems of equation to triangular matrix or trigonalisables, which is not the case of the implicit schemes.

The methods of resolution of the diagrams can call upon methods of optimization as to traditional algebraic methods.

Numerical example of diagram

Let us leave the following equation: $\backslash \; forall\; X\; \backslash \; in\; 1,\; u\text{'}\; (X)\; -\; \backslash \; tau\; U\; (X)\; =\; 0,\; U\; (0)\; =\; u\_0$

Then, one chooses to write the diagram of order 1 of the derived first in all the points of a grid with constant step $\backslash \; \{x\_0\; =\; 0,\; x\_1\; =\; H,\; x\_2\; =\; 2:00,\dots ,\; x\_M\; =\; M\; H\; =\; 1\; \backslash \}$. One seeks $M$ exactly unknown, the values which one will write $u\_n\; =\; U\; (N\; H),\; \backslash \; forall\; N\; \backslash \; in\; \backslash \; \{1,\dots ,\; M\; \backslash \}$. The diagram is then called explicit diagram of Euler of order 1:

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; \{0,\dots ,\; M-1\; \backslash \},\; \backslash \; frac\; \{u\_\; \{N\; +\; 1\}\; -\; u\_n\}\; \{H\}\; -\; \backslash \; tau\; u\_n\; =\; 0$

What explicitly gives the relation of recurrence between $u\_n$ and its successor $u\_\; \{N\; +\; 1\}$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; \{0,\dots ,\; M-1\; \backslash \},\; u\_\; \{N\; +\; 1\}\; =\; (1\; +\; H\; \backslash \; tau)\; u\_n$

That is to say a geometrical continuation which gives us rather easily, by replacing $h$ by its value $\backslash \; frac\; \{1\}\; \{M\}$:

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; \{0,\dots ,\; M-1\; \backslash \},\; u\_n\; =\; \backslash \; left\; (1\; +\; \backslash \; frac\; \{\backslash \; tau\}\; \{M\}\; \backslash \; right)\; ^n\; u\_0$

One thus lays out for all the points of the grid of the value of the solution of the problem according to the method the finite differences. For the points not contained on the grid, it will then be necessary to make an assumption on the quality of the solution, for example to suppose that the function is constant or closely connected per pieces.

Rather let us choose to leave the diagram of order 2 of the derived first, except for the point $n\; =\; 1$ for which one takes again the diagram of order 1:

$\backslash \; frac\; \{u\_1\; -\; u\_0\}\; \{H\}\; -\; \backslash \; tau\; u\_0\; =\; 0$

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; \{1,\dots ,\; M-1\; \backslash \},\; \backslash \; frac\; \{u\_\; \{N\; +\; 1\}\; -\; u\_\; \{N\; -\; 1\}\}\; \{2:00\}\; -\; \backslash \; tau\; u\_n\; =\; 0$

This gives then us

$A\; \backslash \; begin\; \{pmatrix\}\; u\_2\; \backslash \; \backslash \; u\_3\; \backslash \; \backslash \dots \; \backslash \; \backslash \; u\_M\; \backslash \; end\; \{pmatrix\}\; =\; \backslash \; begin\; \{pmatrix\}\; (1+\; \backslash \; tau\; H)\; u\_0\; \backslash \; \backslash \; 0\; \backslash \; \backslash \dots \; \backslash \; \backslash \; 0\; \backslash \; end\; \{pmatrix\}$

where the matrix $A$ is tridiagonale, with $-2\; \backslash \; tau\; h$ on the diagonal, of the 1 on the 1st surdiagonale and the -1 on the 1st sousdiagonale.

Convergences

The second concept supplements the first. It treats convergence of a numerical diagram. Indeed, modeling by differences finished is couple (conditions (initial, final, etc), diagram numerical) which one does not know a priori not if its possible solution near or not to a real solution is hoped of the initial system (conditions (initial, final, etc), differential equations). To speak about convergence, it is necessary to include/understand that one according to a criterion similar to those, if one studies a function $f$ on an interval $$:

- simple convergence: in any point the evaluated approximation tightens towards the true value of the solution when the step tends towards 0

- convergence normalizes 2 of them $\backslash \; left\; (\; ||F||\_2\; =\; \backslash \; sqrt\; \{\backslash \; int\_a^b\}\; \backslash \; right)$: this standard tends towards 0 with the step

- absolute convergence or in absolute standard $\backslash \; left\; (\; ||F||\_\; \backslash \; infty\; =\; \backslash \; max\_\; \{T\; \backslash \; in\}\; \backslash \; right)$: this standard tends towards 0 with the step

One must then study the quality of the convergence of this numerical diagram according to criteria such as stability, the robustness and well-sure various standards (2 or $\backslash \; infty$ most of the time). That can as well depend on the method of discretization, of the nature of the equations that initial conditions (according to the nature of the problem arising).

- Stability, the robustness of the diagram: to add

Example of error of approximation

For a given grid, the error between the approximate solution and the real solution are determined by the truncation error, as in particular definite in the theorem of Taylor with integral remainder or within the meaning of the restriction on a finished part of a Série of Taylor which has an infinite number of terms.

There still, the error depends on what one intends to measure. The error can indeed be measured point by point (simple convergence) or according to a standard 2 or infinite.

In the last example presented, one knows the exact solution of the equation, which is an exponential function.

$u\; (X)\; =\; u\_0\; e^\; \{(\backslash \; tau\; X)\}$

One can thus evaluate the difference directly between the real version and the version estimated by approximation for a grid $M$ intervals:

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; \{0,\dots ,\; M\; \backslash \},\; U\; (x\_n)\; -\; u\_n\; =\; u\_0\; \backslash \; left\; e^\; \{\backslash \; frac\; \{\backslash \; tau\; N\}\; \{M\}\}\; -\; \backslash \; left\; (1\; +\; \backslash \; frac\; \{\backslash \; tau\}\; \{M\}\; \backslash \; right)\; ^n\; \backslash \; right$

Here, there is an exact measurement of the error point by point. Note that one can as to show as if the grid is refined infinitely (i.e. the number of points $M$ tends towards the infinite one), then simple convergence is assured. Let us take an example: the point $\backslash \; frac\; \{1\}\; \{2\}$. Let us distinguish the even case $M$. The case where $M$ is odd is identical by framing, even if it is a little more complicated.

If M is even, then the point $x\_\; \{\backslash \; frac\; \{M\}\; \{2\}\}\; =\; \backslash \; frac\; \{1\}\; \{2\}$. The difference in image is then

$u\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{2\}\; \backslash \; right)\; -\; u\_\; \{\backslash \; frac\; \{M\}\; \{2\}\}\; =\; u\_0\; \backslash \; left\; e^\; \{\backslash \; frac\; \{\backslash \; tau\}\; \{2\}\}\; -\; \backslash \; left\; (1\; +\; \backslash \; frac\; \{\backslash \; tau\}\; \{M\}\; \backslash \; right)\; ^\; \{M/2\}\; \backslash \; right$

However, a traditional result of analysis gives us the limit of the continuation $\backslash \; left\; (1\; +\; \backslash \; frac\; \{\backslash \; tau\}\; \{M\}\; \backslash \; right)\; ^\; \{M/2\}$ is $e^\; \{\backslash \; frac\; \{\backslash \; tau\}\; \{2\}\}$, with the result that the difference $u\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{2\}\; \backslash \; right)\; -\; u\_\; \{\backslash \; frac\; \{M\}\; \{2\}\}$ tends well towards 0.

It is in fact possible, by using the character archimédien and the continuity of the solutions, to show that simple convergence is everywhere assured…

See too

Related articles

• Consistency of a numerical diagram
• Finite element method
• Method of finished volumes
• Theorem of Lax
• Theorem of Lax-Wendroff

External bonds and documents

• Bonds towards tools of discretization (downloadable)

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