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Numerical methods and course review · Differential Equations | Math Infinitum

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Differential Equations
Numerical methods and course review

Differential Equations · Chapter 10

Numerical methods and course review

When no formula exists, an initial value problem can still be run forward, step by step along the slopes the equation gives. Euler’s method and its error, the better methods of Heun and of Runge and Kutta, how to choose a step, and what numbers can and cannot say — then a review of the whole course.

When no formula exists, an initial value problem can still be run forward, step by step along the slopes the equation gives. Euler’s method and its error, the better methods of Heun and of Runge and Kutta, how to choose a step, and what numbers can and cannot say — then a review of the whole course.

y_{n + 1} = y_{n} + h f (t_{n}, y_{n})

t_{n + 1} = t_{n} + h

h > 0

O (h^{4})

k_{1} = f (t_{n}, y_{n})

y_{n} \approx y (t_{n})

O (h^{2})

h ⟶ 0

Lessons

Build the foundation

01Why solve numerically?Most differential equations have no formula for their solutions. Even one as short as y′ = x − y² has none in the functions of calculus. Yet the existence and uniqueness theorem says its initial value problem has exactly one solution, so there is something definite to find. A numerical method does not look for a formula. It computes approximate values of that one solution at a chain of points a small step apart, starting from the initial value and using nothing but the slope the equation gives.Read lesson →
02Euler’s methodThe direction field already says how to walk. Stand at the starting point, read the slope the equation gives there, and follow it in a straight line for one step. At the new point read the slope again, turn, and take the next step. The path is a broken line of straight pieces, each starting off tangent to a solution. This is Euler’s method, the simplest way to run a differential equation forward, and every better method refines it.Read lesson →
03The error of Euler’s methodEach Euler step follows a tangent, and the solution bends away from its tangent. Over one step the gap grows with the square of the step: halve the step and the gap is a quarter. But crossing an interval takes more steps when they are shorter — twice as many at half the size — so the gaps add up to something proportional to the step itself. Halve h and the final error halves. That is what it means for Euler’s method to be of order one, and it is why it is slow: ten times the accuracy costs ten times the work.Read lesson →
04Heun’s methodEuler’s method uses the slope at the start of a step as if it held all the way across, and it does not: by the end of the step the slope has changed. Heun’s idea is to take the Euler step only as a trial, read the slope where it lands, and then take the real step with the average of the two slopes. The average is much closer to the true mean slope over the step, and the error falls with the square of the step: halve h and the error is a quarter.

Read lesson →

05The Runge–Kutta methodIf two slopes averaged give order two, more slopes, well placed, give more. The classical Runge–Kutta method reads the slope four times in each step: at the start, twice in the middle, and at the end, each reading taken at a point reached with the one before. Their weighted average, with the middle counted twice, is accurate to order four: halve the step and the error falls sixteenfold. For decades it was the method, and it is still the first one to reach for.Read lesson →

06Choosing the step sizeA step can be too long in two ways. It can be too long for accuracy, and halving it and comparing the two answers shows by how much. And it can be too long for the method to behave at all: on a solution that dies away fast, Euler’s method with a long step overshoots zero, overshoots back further, and the numbers explode while the true solution quietly vanishes. Too short a step has a cost as well: more work, and more rounding in the arithmetic. Good software therefore changes the step as it goes, short where the solution changes quickly and long where it is smooth.Read lesson →

07Formulas and numbersA formula and a table of numbers answer different questions. A formula holds every solution at once, shows how the answer depends on the constants and the parameters, and says what happens as x grows without limit. Numbers answer one initial value problem, for one set of parameters, over one finite stretch — but they answer it for almost any equation. The two check each other: a formula is tested against numbers, and a program against equations whose solutions are known. And some questions only analysis can answer, because numbers never warn that a solution has stopped existing, or that it is not the only one.Read lesson →

08Course reviewThe whole course in one sitting. Classify an equation first, because the classification picks the method: separable, linear or exact for first order; the characteristic equation for constant coefficients; a guess, variation of parameters or the Laplace transform for forcing; eigenvalues for a linear system and the Jacobian for a nonlinear one; a series when the coefficients vary; and a numerical method when nothing else applies. Then read what the answer says, and check it against the direction field or the phase portrait.Read lesson →