Method of the characteristics

In Mathematical, the method of the characteristics is a technique making it possible to solve the partial derivative equations. Particularly adapted to the problems of transport, it is used in many fields such as the Mécanique of the fluids or the transport of particles.

In certain particular cases, the method of the characteristics can allow the purely analytical resolution of the EDP. Dand the more complex cases (met for example in modeling of the physical systems), the method of the characteristics can be used like a method of resolution numerical of the problem.

Analytical method

General principle

For a partial derivative equation (EDP) first order, the method of the characteristics seeks curves (called “characteristic lines”, or more simply “characteristic”) the length whose the EDP is reduced to simple a ordinary differential equation (EDO). The resolution of the EDO along a characteristic makes it possible to find the solution of the original problem.

Example: Analytical resolution of the transport equation

A function $u$ is sought:

$u:\; (X,\; T)\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; \backslash \; times\; \backslash \; mathbb\; \{R\}\; ^+\; \backslash \; mapsto\; U\; (X,\; T)\; \backslash \; in\; \backslash \; mathbb\; \{R\}$

solution of the following problem:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{array\}\; \{L\; L\}$

\ displaystyle \ frac {\ partial U} {\ partial T} + C \ frac {\ partial U} {\ partial X} = 0, \ quad & (X, T) \ in \ mathbb {R} \ times \ mathbb {R} ^+ \ \ U (X, 0) = u_0 (X), & X \ in \ mathbb {R} \end{array}\right. in which $\backslash \; textstyle\; c$ is a constant.

One seeks a characteristic line $\backslash \; textstyle\; \backslash \; left\; (X\; (S),\; T\; (S)\; \backslash \; right)$ along which this first order EDP would be reduced to a EDO. Let us calculate the derivative of $u$ along such a curve:

$$

\ frac {of the} {ds} = \ frac {dt} {ds} \, \ frac {\ partial U} {\ partial T} + \ frac {dx} {ds} \, \ frac {\ partial U} {\ partial X}

It will be noticed easily that by imposing $\backslash \; textstyle\; \backslash \; frac\; \{dt\}\; \{ds\}\; =\; 1$ and $\backslash \; textstyle\; \backslash \; frac\; \{dx\}\; \{ds\}\; =\; c$, one obtains: $\backslash \; textstyle\; \backslash \; frac\; \{of\; the\}\; \{ds\}\; =\; \backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; T\}\; +\; C\; \backslash \; frac\; \{\backslash \; partial\; U\}\; \{\backslash \; partial\; X\}\; =\; 0$. the solution of the equation thus remains constant along the characteristic line.

We thus find ourselves with three ordinary differential equations to solve:

• $\backslash \; frac\; \{dt\}\; \{ds\}\; =\; 1$: by posing $\backslash \; textstyle\; T\; (0)\; =0$, one obtains:

$\backslash \; forall\; S\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; ^+,\; \backslash \; quad\; T\; (S)\; =s$

• $\backslash \; frac\; \{dx\}\; \{ds\}\; =\; c$: by noting $\backslash \; textstyle\; X\; (0)\; =x\_0$, one obtains:

$\backslash \; forall\; S\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; ^+,\; \backslash \; quad\; X\; (S)\; =x\_0+c\; \backslash ,\; s=x\_0+c\; \backslash ,\; t$

• $\backslash \; frac\; \{of\; the\}\; \{ds\}\; =\; 0$:

$\backslash \; forall\; S\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; ^+,\; \backslash \; quad\; U\; (S)\; =\; U\; (0)\; =\; U\; (x\_0,0)\; =\; u\_0\; (x\_0)$

In this case, the characteristic lines are thus lines of slope $\backslash \; textstyle\; c$, the length whose the solution remains constant. The value of the solution in a point $\backslash \; textstyle\; (X,\; T)\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; \backslash \; times\; \backslash \; mathbb\; \{R\}\; ^+$ can thus be found by seeking the value of the initial condition $\backslash \; textstyle\; u\_0$ in the beginning $\backslash \; textstyle\; x\_0=x-c\; \backslash ,\; t$ of the characteristic line:

$\backslash \; forall\; (X,\; T)\; \backslash \; in\; \backslash \; mathbb\; \{R\}\; \backslash \; times\; \backslash \; mathbb\; \{R\}\; ^+,\; \backslash \; quad\; U\; (X,\; T)\; =\; u\_0\; (x-c\; \backslash ,\; T)$

Numerical method

General principle

Example

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