Method of characteristics

From Academic Kids

In mathematics, the method of characteristics is a way of solving partial differential equations, and systems of PDEs.

For a first-order PDE, a characteristic is a line in phase space (comprising the independents, the dependent, and the partial derivatives of the dependent with respect to the independents) along which the PDE degenerates into an ordinary differential equation.

In two dimensions, even a non-linear first-order PDE can always be written in the form

<math>F(x, y, u, u_x, u_y) = 0,<math>

where x and y are the independents, u(x,y) is the unknown solution, and the remaining arguments are the partial derivatives of the solution u. In the method of characteristics the contour map of F, comprising the level curves where F is constant, is used to recover the solution. This procedure is exact as long as the solution is smooth and differentiable.

While level curves do not intersect, characteristics sometimes do. When this happens the solution function is multivalued, the correct branch must be selected, and where they meet discontinuities arise in the form of shockwaves. Characteristics may also fail to cover part of the domain of the PDE: This is called a rarefaction, and a solution typically exists only in a weak, i.e., integral equation, sense.

Example

Consider the one-dimensional scalar conservation equation

<math> u_t + f_x(u)=0. <math>

Here u and f are scalar, with u(x,t) a function of x and t.

By the chain rule this equation implies that

<math> u_t + f_u u_x = 0 <math>

Consider a curve in the x-t plane <math>\eta(t)<math> parameterized as a function of t so that <math>\eta(0)=x_0 <math> for some <math>x_0<math> and <math> \eta_t = f_u <math>

<math> \frac{d(u)}{dt} = u_t + \eta_t u_\eta = u_t + f_u u_\eta = 0<math>
<math>\int_0^t \frac{d(u)}{dt} dt = \int_0^t 0 dt<math>
<math>\int_0^t du = 0<math>
<math>u(\eta(t),t)-u(\eta(0),0)= 0<math>
<math>u(\eta(t),t)=u(x_0,0)<math>

So anywhere along the curve <math> \eta(t) <math> (the characteristic curve) the function is the same as at the beginning.

A similar analysis shows that the curve is in fact a straight line for this case.

Bibliography

  • A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002.
  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.
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