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In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation using various techniques, but it is not so easy to do the same for PDE systems.

Let's say we have first order, strictly hyperbolic, nonlinear, coupled 2x2 system of pdes, in one dimension, that looks like this: $$ \begin{cases} u_t+f(u,v) u_x=h(u,v) \\[2ex] v_t+g(u,v) v_x=m(u,v) \\[2ex] (u(x,0),v(x,0))=(u_0 (x), v_0 (x)) \end{cases} $$ (This is a Cauchy problem). If $(u_0 (x), v_0 (x))$ is discontinuous, e.g. $$(u(x,0),v(x,0))= \begin{cases} (u_l , v_l), x<0 \\[2ex] (u_r , v_r), x\geq0 \end{cases}$$ we would talk about Riemann problem.

I want to solve this problem using method of characteristics.

My question is: Is there any book/note/paper where I could find a step by step solution of quasilinear system that looks like this by using method of characteristics? Or maybe an easier quasilinear system (e.g. if we put right hand sides equal zero, put $f$ equal to $g$,...). It doesn't matter if it is a Riemann or Cauchy problem. I just want to see how system of this type could be solved in detail from start to finish. It could be shown on any concrete system (and not for the general case I've written above).

In the literature I've found something in the book:

Toro - Riemann solvers and numerical methods of fluid dynamics, 2009,

but it is very short. Additionaly, I have found a similar questions on Mathoverflow (question1 and question2).

If anyone has a lot of time he/she could write this all down on a concrete example but the reference in the literature would be just fine.

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    $\begingroup$ As far as I know, the method of characteristics works only for a single first order PDE and, in general, not for systems. $\endgroup$
    – Deane Yang
    Commented Jul 19, 2018 at 15:41
  • $\begingroup$ Essentially, in the method of characteristics a first-order PDE tells you how the dependent variable is changing in a certain direction, and you put those directions together to make the characteristic curves. But for a system, the direction for one variable will generally be different than the direction for another variable. $\endgroup$
    – Robert Israel
    Commented Jul 19, 2018 at 17:33
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    $\begingroup$ @DeaneYang: you are right in general, when the domain manifold can have arbitrary number of dimensions. The case of two independent variables and two unknowns is sort of a special case; in the homogeneous setting it was solved by Riemann. $\endgroup$
    – Willie Wong
    Commented Jul 19, 2018 at 21:34
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    $\begingroup$ In fact, the method of characteristics can indeed be used to solve many nonlinear, first-order $2$-by-$2$ systems using only ODE methods, as was discovered in the 19th century and was developed by Darboux in an important memoir in 1870. It now goes by the name Darboux' Method (in French, la methode de Darboux and so rendered in English as the method of Darboux in many places). However, the generic system is not explicitly solvable by Darboux' Method, as was shown by Lie in particular cases, with the general theorem proved by Goursat some years later (but, in any case, by 1896). $\endgroup$
    – Robert Bryant
    Commented Dec 17, 2018 at 12:50
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    $\begingroup$ A method for integrating simultaneous first order PDE in two dependent and two independent variables was developed by Hamburger in 1876. (It extends the method of characteristics to this case.) The method is explained in the fifth volume of A.R. Forsyth's Theory of Differential Equations (1890-1906), widely available via free download. Hamburger's method was extended by M. Kourensky in "A Method of Integrating the General form of a System of Partial Differential Equations of the First Order in two Dependent and two Independent Variables" Proc. London Math. Soc..1930; s2-31: 407-416. $\endgroup$
    – Phil Harmsworth
    Commented Dec 26, 2020 at 9:15

2 Answers 2

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When the RHS is 0, you are basically asking about the method of Riemann invariants. A quick summary of the method you can find in the second section of

Lax (1964), J. Math. Phys.

Alternatively, I am pretty sure it is also discussed somewhere in the second volume of Courant and Hilbert.

EDIT: I see that there's also a decent write-up on Wikipedia, you just need to know where to look.

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  • $\begingroup$ Thank you for the answer. Sorry for a very late reply. Lax paper is great. Section 2, Quasilinear systems for two unknowns, explains it beautifully. Only drawback of this method is the requirement that the system is genuinely nonlinear (a lot 2x2 systems do not satisfy this). Courant, Hilbert, 1989 book is very good but it usually deals with linear and semilinear systems (quasilinear and generally nonlinear systems are mentioned a lot especially in the Chapter V but not in much details - I maybe missed something, it's a big book). Wikipedia (Riemann invariant page) is good for the basics. $\endgroup$
    – Mark
    Commented Dec 17, 2018 at 11:30
  • $\begingroup$ @Mark: the genuine nonlinear condition is ONLY required if you want to prove the existence of shocks. The basic existence theorem and the basic method of characteristics do not require the genuine nonlinear condition. $\endgroup$
    – Willie Wong
    Commented Dec 17, 2018 at 13:59
  • $\begingroup$ Thank you for the addition. These days I mostly deal with shocks, so this is good to know. $\endgroup$
    – Mark
    Commented Dec 17, 2018 at 14:20
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The method of characteristics for $2\times2$ systems is discussed by C. Dafermos in his book. In the third edition, it is Chapter XII.

However, I want to make a few points:

  • as mentionned by Willie, it was elaborated by P. Lax, who used it to prove that genuine nonlinearity implies the blow-up of first derivatives in finite time, for very general initial data.
  • The method is of little help for applications, because we don't want to be llimited by the onset of shock waves.
  • Lax calculus has been generalized to $n\times n$ systems of the rich class. A system is rich if it can be put in a conservative form and it can also be put in a diagonal form. This is called semi-hamiltonian system by the russian school.
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  • $\begingroup$ I usually use method of characteristics for the equations, so I was surprised how different and more complicated is the method of generalized characteristics given in Chapter XII of Dafermos book. The assumption there is that the both families are genuine nonlinear. I strugle to understand the importance of genuine nonlinearity. For example, if genuine nonlinearity of both families for 2x2 system, implies blow-up of the first derivatives, what would happen if the first family is linearly degenerate and the second one is genuine nonlinear? Or if both families are linearly degenerate? $\endgroup$
    – Mark
    Commented Dec 23, 2018 at 17:40
  • $\begingroup$ About systems of rich class: that would be the shallow water system for example? It is given in divergent conservative form in $(\rho,u)$ coordinates and it can be transformed in a diagonal form (in Riemann invariants). Or the rich class system can be also be any 2x2 system that is given in the non-divergence form and that can be transformed into the diagonal one? I think I could use systems of rich class in one of the problems I am working on for some time, so I would be really grateful if you can recommend me some book(paper) where I could find more about them. And thank you for your answer. $\endgroup$
    – Mark
    Commented Dec 23, 2018 at 17:47
  • $\begingroup$ @Mark. You could also have a look to the second volume of my book on Hyperbolic conservation laws. Rich systems are those for which compensated compactness can be applied. Genuine nonlinearity is the property that prevent oscillations to accumulate. Systems with a linearly degenerate field (the opposite of GNL) are always more difficult to handle. $\endgroup$
    – Denis Serre
    Commented Dec 23, 2018 at 17:58
  • $\begingroup$ Thank you for the follow up. I will check out your book (I already use first volume from time to time). Do you have any concrete example of a system where I could see why is it more difficult to handle linearly degenerate case than GNL case? $\endgroup$
    – Mark
    Commented Dec 24, 2018 at 8:45
  • $\begingroup$ @Mark. The so-called Born-Infeld system ($w_+zw_x=0$ and $z_t+wz_x=0$) has linearly degenerate fields. If the initial data is periodic, then the solution, already space-periodic, admits a second period in the space-time plane. If you rescale by $(x,t)\mapsto(x/\epsilon,t/\epsilon)$, you obtain a weakly converging sequence, which does not converge strongly ; this cannot happen for a genuinely nonlinear system. $\endgroup$
    – Denis Serre
    Commented Dec 24, 2018 at 10:26

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