Method of characteristics
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface. Characteristics of first-order partial differential equation. For a first-order PDE (partial differential equation), the method of characteristics discovers curves (called characteristic curves or just characteristics) along which  the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. For the sake of simplicity, we confine our attention to the case of a function of two independent variables "x" and "y" for the moment. Consider a quasilinear PDE of the form Suppose that a solution "z" is known, and consider the surface graph "z" = "z"("x","y") in R. A normal vector to this surface is given by As a result, equation () is equivalent to the geometrical statement that the  vector field is tangent to the surface "z" = "z"("x","y") at every point, for the dot product of this vector field with the above normal vector is zero. In other words, the graph of the solution must be a union of integral curves of this vector field. These integral curves are called the characteristic curves of the original partial differential equation and are given by the "Lagrange–Charpit equations" A parametrization invariant form of the "Lagrange–Charpit equations" is: Linear and quasilinear cases. Consider now a PDE of the form For this PDE to be linear, the coefficients "a" may  be functions of the spatial variables only, and independent of "u". For it to be quasilinear, "a" may also depend on the value of the function, but not on any derivatives. The distinction between these two cases is inessential for the discussion here. For a linear or quasilinear PDE, the characteristic curves are given parametrically by such that the following system of ODEs is satisfied Equations () and () give the characteristics of the PDE. Proof for quasilinear Case. In the quasilinear case, the use of the method of characteristics is justified by Grönwall's inequality. The above equation may be written  as formula_8 We must distinguish between the solutions to the ODE and the solutions to the PDE, which we do not know are equal "a priori." Letting capital letters be the solutions to the ODE we find formula_9 formula_10 Examining formula_11, we find, upon differentiating that formula_12 which is the same as formula_13 We cannot conclude the above is 0 as we would like, since the PDE only guarantees us that this relationship is satisfied for formula_14, formula_15, and we do not yet know that formula_16. However, we can see that formula_17 since by the PDE, the last term  is 0. This equals formula_18 By the triangle inequality, we have formula_19 Assuming formula_20 are at least formula_21, we can bound this for small times. Choose a neighborhood formula_22 around formula_23 small enough such that formula_20 are locally Lipschitz. By continuity, formula_25 will remain in formula_22 for small enough formula_27. Since formula_28, we also have that formula_29 will be in formula_22 for small enough formula_31 by continuity. So, formula_32 and formula_33 for formula_34. Additionally, formula_35 for some formula_36 for formula_34 by compactness. From this, we find the above is bounded as formula_38 for some formula_39. It is a straightforward application  of Grönwall's Inequality to show that since formula_40 we have formula_41 for as long as this inequality holds. We have some interval formula_42 such that formula_43 in this interval. Choose the largest formula_44 such that this is true. Then, by continuity, formula_45. Provided the ODE still has a solution in some interval after formula_44, we can repeat the argument above to find that formula_43 in a larger interval. Thus, so long as the ODE has a solution, we have formula_43. Fully nonlinear case. Consider the partial differential equation where the variables "p" are shorthand for the partial derivatives Let  ("x"("s"),"u"("s"),"p"("s")) be a curve in R. Suppose that "u" is any solution, and that Along a solution, differentiating () with respect to "s" gives The second equation follows from applying the chain rule to a solution "u", and the third follows by taking an exterior derivative of the relation formula_54. Manipulating these equations gives where λ is a constant. Writing these equations more symmetrically, one obtains the Lagrange–Charpit equations for the characteristic Geometrically, the method of characteristics in the fully nonlinear case can be interpreted as requiring that the Monge cone of the differential equation should everywhere be tangent to  the graph of the solution. Example. As an example, consider the advection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs). where formula_58 is constant and formula_59 is a function of formula_60 and formula_61. We want to transform this linear first-order PDE into an ODE along the appropriate curve; i.e. something of the form where formula_63 is a characteristic line. First, we find by the chain rule. Now, if we set formula_65 and formula_66 we get which is the left hand side of the PDE we started with. Thus So, along the characteristic line formula_69 , the original PDE becomes the ODE formula_70. That is to say that along the characteristics, the solution is constant. Thus, formula_71 where formula_72 and formula_73 lie on the same characteristic. Therefore, to determine the general solution, it is enough to find the characteristics by solving the characteristic system of ODEs: In this case, the characteristic lines are straight lines with slope formula_58, and the value of formula_59 remains constant along any characteristic line. Characteristics of linear differential operators. Let "X" be a differentiable manifold and "P" a linear differential operator of order "k". In a local coordinate system "x", in  which "α" denotes a multi-index. The principal symbol of "P", denoted "σ", is the function on the cotangent bundle T"X" defined in these local coordinates by where the "ξ" are the fiber coordinates on the cotangent bundle induced by the coordinate differentials "dx". Although this is defined using a particular coordinate system, the transformation law relating the "ξ" and the "x" ensures that "σ" is a well-defined function on the cotangent bundle. The function "σ" is homogeneous of degree "k" in the "ξ" variable. The zeros of "σ", away from the zero section of T"X", are the characteristics  of "P". A hypersurface of "X" defined by the equation "F"("x") = "c" is called a characteristic hypersurface at "x" if Invariantly, a characteristic hypersurface is a hypersurface whose conormal bundle is in the characteristic set of "P". Qualitative analysis of characteristics. Characteristics are also a powerful tool for gaining qualitative insight into a PDE. One can use the crossings of the characteristics to find shock waves for potential flow in a compressible fluid. Intuitively, we can think of each characteristic line implying a solution to formula_59 along itself. Thus, when two characteristics cross, the function becomes multi-valued resulting  in a non-physical solution. Physically, this contradiction is removed by the formation of a shock wave, a tangential discontinuity or a weak discontinuity and can result in non-potential flow, violating the initial assumptions. Characteristics may fail to cover part of the domain of the PDE. This is called a rarefaction, and indicates the solution typically exists only in a weak, i.e. integral equation, sense. The direction of the characteristic lines indicate the flow of values through the solution, as the example above demonstrates. This kind of knowledge is useful when solving PDEs numerically as it can indicate which  finite difference scheme is best for the problem.