Method of characteristics

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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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x, y, u, u_x, u_y) = 0,}

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.

The first order wave equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = 0 \!}

describes the movement of a wave in one direction with no change of shape. A solution is shown in Figure 1.3 below as a surface plot and a contour plot.

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Example

Consider the one-dimensional scalar conservation equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_t + f_x(u)=0. }

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

By the chain rule this equation implies that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_t + f_u u_x = 0 }

Consider a curve in the x-t plane Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta(t)} parameterized as a function of t so that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta(0)=x_0 } for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_t = f_u }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d(u)}{dt}\,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = u_t + \eta_t u_\eta\,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = u_t + f_u u_\eta\,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 0\,}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^t \frac{d(u)}{dt} dt = \int_0^t 0 \, dt}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^t \, du = 0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u[\eta(t),t]-u[\eta(0),0]= 0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u[\eta(t),t]=u(x_0,0)}

So anywhere along the curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta(t) } (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

  • L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
  • A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9