The Method of Characteristics

The method of characteristics turns a partial differential equation into an ordinary differential equation along carefully chosen paths. We will build that idea from a moving shape, then follow it until those paths collide. Let's start with water. A long, narrow tank of it, seen from the side, and something has disturbed the surface: a swell, sitting here. Now pick a spot along the tank (any spot) and ask one very small question. How high is the water, right there? Down here it is barely raised. Walk toward the swell and it is higher. Keep going, up to the top, higher still, and past that it drops away again. So every position hands you back one number. That is all a function is, and this one has a name: u. Except water does not hold still, and that ruins it. Watch. And I owe you an explanation for what you are about to see, because the swell is going to keep its shape exactly. That is not a law of water: it is the simplest case, and the one to understand first. Picture the whole tank drifting downstream at a steady speed, carrying whatever sits on it the way a river carries a leaf. Nothing spreads, nothing dies. It just goes. Now stop moving, and let it come to you instead. Stand at this one place. The water climbs past you, and then it sinks away again, and you never took a step. So knowing where you are is no longer enough to know how high the water is. You have to say when as well. u of x and t. Two inputs means two ways to change it, so let us change them one at a time. First, freeze the clock. Nothing moves; this is one instant, held. Stand here, and walk this far along the tank. You changed x by that much, and look: u changed too, by that much. Same instant, different place, different height. Take a shorter walk and u changes less. A longer one, and it changes more. There is a rate hiding in there (height gained per step of x) and that rate is what u sub x means. Now the other one. Put x back where it was and leave it there: one place, this stake, and you do not move again. On the right I will keep a record of the height at that one stake, against time. Not against position; the position is fixed now. Against time. Watch both dots. The swell arrives, the water climbs the stake, and the record climbs with it. The crest passes, and the two of them come back down together. One number, drawn twice. You never moved, and u changed anyway. So there is a second rate (height gained per tick of the clock) and that one is u sub t. Two rates, then. And they are not independent: here is why. Here is the surface at one instant, and here it is a moment later. Nothing about the shape changed. Every part of it simply moved to the right, by the speed c times that moment. So the height at your feet now is not a new number at all. It is the height that used to sit a little way upstream (exactly c delta t upstream) and has since arrived. Now read that as a rate. The water rises under you at whatever rate the shape slopes downward, multiplied by how fast the shape is coming. Steeper slope, or faster water, and the height changes quicker. Tidy it up, and there is the transport equation. Every symbol in it is something we pointed at first. And now the reason anyone would want to solve it. This is a rule about rates, here and now. A solution is the height at every place at every time, including times that have not happened yet. Solving it means predicting the water. And there is more than one way to do that. You could guess that the shape merely travels and verify it; we have very nearly done that already. You could break the profile into sine waves and march each one forward. Or you could hunt for paths through the x t plane along which u is not allowed to change at all. That last one is the method of characteristics. It is one method among several, but it is the one that turns a partial differential equation into an ordinary one, and that is a trade worth understanding.

Let's try something that looks like a trick, and turns out to be the whole method. Instead of asking what u does everywhere at once, we are going to choose a curve with x prime of t equal to c, then walk across the x t plane and watch u as we go. Here is that curve, and here is where we stand on it at time t: at the position x of t, at height t. Now give the value of u at that moving point a name. Call it z of t. And z is a function of one variable. One variable means we can simply differentiate it, and the chain rule tells us exactly what comes out. u sub x times x prime of t, coming through the x slot, plus u sub t, coming through the t slot. Nothing clever has happened yet. But look hard at what is sitting there: u sub x, u sub t, and a factor x prime of t that nobody has told us how to choose. So let's choose it. Set x prime of t equal to c, the speed out of the equation. And now read the right hand side. u sub t plus c u sub x. That is precisely the left hand side of our equation, and our equation says it is zero. So z prime is zero. The value of u does not change at all as we walk along that curve. And which curve is it? x prime equals c says that x is x naught plus c t. A straight line of slope c in the x t plane. That line has a name: it is called a characteristic. Every equation of this kind has a whole family of them, one through each starting point. And the partial differential equation, which was a statement about two derivatives at once, has become a statement about one derivative along a line.

We already have the family of parallel characteristics. Keep the initial profile f in view on the right. Every point of the x t plane sits on exactly one of those paths, so one path is enough to recover its value. Now pick any point x, t, up here. Follow its characteristic back down to time zero. It lands on the horizontal axis at x minus c t. And u is constant all the way along that line. So the value up at x, t is the value down at the foot, which is nothing but the initial data f, evaluated at x minus c t. And there is our solution. That is the whole answer. No integrals, no infinite series, no separation of variables. Whatever shape you start with, you get that same shape back, only shifted. Watch what that means. Here is f, the blob sitting there at time zero. A moment later, the whole graph has picked itself up and moved to the right. Not one thing about its shape has changed. Later still, further along, and it is still exactly the same blob. That is the only thing this equation ever does. And here is one particle of dye. On the left it climbs its own characteristic, from that foot up to the point we asked about. On the right it rides the crest along at speed c. Solving this equation was never more than following the flow. One last remark, and it matters in a minute. The data is carried along untouched, so if f has a sharp corner in it, that corner rides along forever and never smooths out. Transport has no memory and no diffusion. It only translates.

Now let's change exactly one thing. Instead of a constant speed c, let the speed depend on time. u sub t plus t u sub x equals zero. Look back at what the derivation actually used. Nothing in it needed c to be constant. Along a curve where u does not change, the chain rule asks x prime of t to equal whatever sits in front of u sub x. Here, that is t. So the characteristics solve x prime equals t. That is an ordinary differential equation, and an easy one. Integrate, and x of t is x naught plus t squared over two. Parabolas. Each one leaves the axis at its own x naught and then takes off. At time zero the slope is zero, so everything starts out standing still. Later they lean over more and more. The dye is accelerating. Now invert it. If I am standing at position x at time t, which parabola am I on? Solve for x naught, and it is x minus t squared over two. And u is constant along each parabola, so u of x, t is f of x minus t squared over two. Same recipe, two lines of work, and the answer is once again the starting profile with a rewritten argument. It is worth checking. Differentiate: u sub t is minus t times f prime, and u sub x is f prime. So u sub t plus t u sub x is minus t f prime, plus t f prime, which is zero. It works.

One last equation, and it is the one that made this method famous. u sub t plus u times u sub x equals zero. The coefficient in front of u sub x is now u itself: the value being carried sets its own speed. Along any curve, the chain rule gives u sub t plus x prime of t times u sub x. Choose x prime of t equal to u, and that derivative becomes the equation's left hand side, which is zero. So u stays constant along the characteristic. A characteristic leaving x naught therefore carries the fixed value f of x naught. Its slope x prime is that same fixed number, so each path is a straight line, but different starting values produce different slopes. Take decreasing initial data. The piece starting at x naught equals zero point five carries the large value one point seven five, so it travels fast. The piece starting at three point five carries only zero point two five, so it barely crawls. The two middle values lie between them. The quick characteristics chase the slow ones ahead. All four paths meet at x equals four, t equals two. They arrive carrying four different values of u, but u is supposed to be a function. At that point it cannot remain smooth and single-valued. The same collision appears in the wave profile. At first the front is smooth. As faster values catch slower ones, it steepens, and steepens again, until the front is nearly vertical. That breaking point is a shock. The smooth solution stops exactly when the first characteristics meet. The method is four steps long, and their order matters. First read the transport speed from the equation. First, write the equation with the two derivatives lined up, and read off the coefficient sitting in front of u sub x. Second, solve x prime of t equals that coefficient. Its solutions are the characteristics, one through each starting point x naught. Third, use the equation to show that u does not change along those curves. Thus u equals f of x naught all the way along. Fourth, invert. Express x naught in terms of x and t, then substitute it into f. That final expression is the solution. Each step is ordinary calculus: read the speed, find its curves, carry the initial value along them, and trace the requested point back to its start. The partial differential equation becomes ordinary along exactly those paths.