# Euler’s method

## Contents

# 6.2. Euler’s method#

Let a first-order initial-value problem be given in the form

We represent a numerical solution of an IVP by its values at a finite collection of nodes, which for now we require to be equally spaced:

The number \(h\) is called the **step size**.

Because we don’t get exactly correct values of the solution at the nodes, we need to take some care with the notation. From now on we let \(\hat{u}(t)\) denote the exact solution of the IVP. The approximate value at \(t_i\) computed at the nodes by our numerical methods will be denoted by \(u_i\approx \hat{u}(t_i)\). Because we are given the initial value \(u(a)=u_0\) exactly, there is no need to distinguish whether we mean \(u_0\) as the exact or the numerical solution.

Consider a piecewise linear interpolant to the (as yet unknown) values \(u_0,u_1,\ldots\), \(u_n\). For \(t_i < t < t_{i+1}\), its slope is

We can connect this derivative to the differential equation by following the model of \(u'=f(t,u)\):

We could view the left-hand side as a forward-difference approximation to \(u'(t)\) at \(t=t_i\). We can rearrange the equation to get **Euler’s method**, our first method for IVPs.

Euler’s method marches ahead in \(t\), obtaining the solution at a new time level explicitly in terms of the latest value.

A basic implementation of Euler’s method is shown in Function 6.2.2. It expects the IVP to be specified as an `ODEProblem`

, as in Demo 6.1.3. The output of Function 6.2.2 is a vector of the nodes and a vector of approximate solution values at those nodes.

**Euler’s method for an initial-value problem**

```
1"""
2 euler(ivp,n)
3
4Apply Euler's method to solve the given IVP using `n` time steps.
5Returns a vector of times and a vector of solution values.
6"""
7function euler(ivp,n)
8 # Time discretization.
9 a,b = ivp.tspan
10 h = (b-a)/n
11 t = [ a + i*h for i in 0:n ]
12
13 # Initial condition and output setup.
14 u = fill(float(ivp.u0),n+1)
15
16 # The time stepping iteration.
17 for i in 1:n
18 u[i+1] = u[i] + h*ivp.f(u[i],ivp.p,t[i])
19 end
20 return t,u
21end
```

About the code

The structure created by `ODEFunction`

contains the data used to define it, and it is accessed in Function 6.2.2 by dot notation, such as `ivp.f`

.

## Local truncation error#

Let \(\hat{u}(t)\) be the exact solution of the IVP (6.2.1), and suppose that somehow we have access to it at \(t=t_i\), so that \(u_i=\hat{u}(t_i)\). How good is \(u_{i+1}\) as an approximation to \(\hat{u}(t_{i+1})\)? The answer is revealed through a Taylor series:

where we used the fact that \(\hat{u}\) satisfies the differential equation.

We now introduce some formalities.

Euler’s method is the particular case of (6.2.5) with \(\phi(t,u,h) = f(t,u)\), but we will see other one-step methods in later sections.

In close analogy with Section 5.5, we define truncation error as the residual of (6.2.5) when the exact solution is inserted.

The following follows immediately from the definitions.

If \(\phi(t,u,0)=f(t,u)\) for any function \(u\), then the method (6.2.5) is consistent.

## Convergence#

While the local truncation error is straightforward to calculate from its definition, it is not the quantity we want to know about and control.

Given an IVP whose exact solution is \(\hat{u}(t)\), the **global error** of approximate solution values \(u_0,u_1,\ldots,u_n\) at times \(t_i\) in (6.2.2) is the vector \([ \hat{u}(t_i) - u_i ]_{\,i=0,\ldots,n}\).

At times the term *global error* may be interpreted as the max-norm of the global error vector, or as its final value.

By our definitions, the local error in stepping from \(t_i\) to \(t_{i+1}\) is \(h\tau_{i+1}(h)\). To reach the time \(t=b\) from \(t=a\) with step size \(h\), we need to take \(n=(b-a)/h\) steps. If we want to reach, say, \(t=(a+b)/2\), then we would have to take \(n/2\) steps, and so on. In fact, to reach any fixed time in the interval, we need to take \(O(n)=O(h^{-1})\) steps. By expressing the local error with a factor of \(h\) taken out, the LTE \(\tau\) itself is accounting for the simple accumulation of error caused by taking \(O(n)\) steps.1

However, global error is not as simple as a sum of local errors. As explained in Theorem 6.1.7 and illustrated in Demo 6.1.8, each step causes a perturbation of the solution that can grow as \(t\) advances. Thus, we have to account for the flow evolution of individual step truncation errors as well as their mere accumulation. That is the subject of the following theorem.

Suppose that the unit local truncation error of the one-step method (6.2.5) satisfies

and that

for all \(t\in[a,b]\), all \(u\), and all \(h>0\). Then the global error satisfies

as \(h\rightarrow 0\).

Define the global error sequence \(ϵ_i=\hat{u}(t_i)-u_i\). Using (6.2.5), we obtain

or

We apply the triangle inequality, (6.2.6), and (6.2.7) to find

The Fundamental Theorem of Calculus implies that

Thus

To get the last line we applied the inequality recursively until reaching \(ϵ_0\), which is zero. Replacing \(i+1\) by \(i\) and simplifying the geometric sum, we get

We observe that \(1+x \le e^x\) for \(x\ge 0\) (see Exercise 5). Hence \((1+hL)^i \le e^{i h L}\), which completes the proof.

The theorem justifies one more general definition.

If the local truncation error of the one-step method (6.2.5) satisfies \(\tau_{i+1}(h)=O(h^p)\) for a positive integer \(p\), then \(p\) is the **order of accuracy** of the formula.

We could restate Theorem 6.2.7 as saying that the global error has the same order of accuracy as the LTE. Note, however, that the \(O(h^p)\) convergence hides a leading constant that grows exponentially in time. When the time interval is bounded as \(h\to 0\), this does not interfere with the conclusion, but the behavior as \(t\to\infty\) contains no such guarantee.

We consider the IVP \(u'=\sin[(u+t)^2]\) over \(0\le t \le 4\), with \(u(0)=-1\).

```
f = (u,p,t) -> sin((t+u)^2);
tspan = (0.0,4.0);
u0 = -1.0;
ivp = ODEProblem(f,u0,tspan)
```

```
ODEProblem with uType Float64 and tType Float64. In-place: false
timespan: (0.0, 4.0)
u0: -1.0
```

Here is the call to Function 6.2.2.

```
t,u = FNC.euler(ivp,20)
plot(t,u,m=2,label="n=20",
xlabel=L"t",ylabel=L"u(t)",title="Solution by Euler's method" )
```

We could define a different interpolant to get a smoother picture above, but the derivation of Euler’s method assumed a piecewise linear interpolant. We can instead request more steps to make the interpolant look smoother.

```
t,u = FNC.euler(ivp,50)
plot!(t,u,m=2,label="n=50")
```

Increasing \(n\) changed the solution noticeably. Since we know that interpolants and finite differences become more accurate as \(h\to 0\), we should anticipate the same behavior from Euler’s method. We don’t have an exact solution to compare to, so we will use a `DifferentialEquations`

solver to construct an accurate reference solution.

```
u_exact = solve(ivp,Tsit5(),reltol=1e-14,abstol=1e-14)
plot!(u_exact,l=(2,:black),label="reference")
```

Now we can perform a convergence study.

```
n = [ round(Int,5*10^k) for k in 0:0.5:3 ]
err = []
for n in n
t,u = FNC.euler(ivp,n)
push!( err, norm(u_exact.(t)-u,Inf) )
end
pretty_table((n=n,err=err),["n","Inf-norm error"])
```

```
┌──────┬────────────────┐
│ n │ Inf-norm error │
├──────┼────────────────┤
│ 5 │ 2.7342 │
│ 16 │ 0.107594 │
│ 50 │ 0.0299962 │
│ 158 │ 0.00885025 │
│ 500 │ 0.00273659 │
│ 1581 │ 0.000859654 │
│ 5000 │ 0.000271243 │
└──────┴────────────────┘
```

The error is approximately cut by a factor of 10 for each increase in \(n\) by the same factor. A log-log plot also confirms first-order convergence. Keep in mind that since \(h=(b-a)/n\), it follows that \(O(h)=O(n^{-1})\).

```
plot(n,err,m=:o,label="results",
xaxis=(:log10,L"n"), yaxis=(:log10,"Inf-norm global error"),
title="Convergence of Euler's method")
# Add line for perfect 1st order.
plot!(n,0.05*(n/n[1]).^(-1),l=:dash,label=L"O(n^{-1})")
```

Euler’s method is the ancestor of the two major families of IVP methods presented in this chapter. Before we describe them, though, we generalize the initial-value problem itself in a crucial way.

## Exercises#

✍ Do two steps of Euler’s method for the following problems using the given step size \(h\). Then, compute the error using the given exact solution.

**(a)**\(u' = -2t u, \ u(0) = 2;\ h=0.1;\ \hat{u}(t) = 2e^{-t^2}\)**(b)**\(u' = u + t, \ u(0) = 2;\ h=0.2;\ \hat{u}(t) = -1-t+3e^t\)**(c)**\(t u' + u = 1, \ u(1) = 6, \ h = 0.25;\ \hat{u}(t) = 1+5/t\)**(d)**\(u' - 2u(1-u) = 0, \ u(0) = 1/2, \ h = 0.25; \ \hat{u}(t) = 1/(1 + e^{-2t})\)⌨ For each IVP, solve the problem using Function 6.2.2. (i) Plot the solution for \(n=320\). (ii) For \(n=10\cdot2^k\), \(k=2,3,\ldots,10\), compute the error at the final time and make a log-log convergence plot, including a reference line for first-order convergence.

**(a)**\(u' = -2t u, \ 0 \le t \le 2, \ u(0) = 2;\ \hat{u}(t) = 2e^{-t^2}\)**(b)**\(u' = u + t, \ 0 \le t \le 1, \ u(0) = 2;\ \hat{u}(t) = -1-t+3e^t\)**(c)**\((1+t^3)uu' = t^2,\ 0 \le xt \le 3, \ u(0) =1;\ \hat{u}(t) = [1+(2/3)\ln (1+xt^3)]^{1/2}\)**(d)**\(u' - 2u(1-u) = 0, \ 0 \le t \le 2, \ u(0) = 1/2; \ \hat{u}(t) = 1/(1 + e^{-2t})\)**(e)**\(v' - (1+x^2) v = 0, \ 1 \le x \le 3, \ v(1) = 1, \ \hat{v}(x) = e^{(x^3+3x-4)/3}\)**(f)**\(v' + (1+x^2) v^2 = 0, \ 0 \le x \le 2, \ v(0) = 2, \ \hat{v}(x) = 6/(2x^3+6x+3)\)**(g)**\(u' = 2(1+t)(1+u^2), \ 0 \le t \le 0.5, \ u(0) = 0, \ \hat{u}(t) = \tan(2t + t^2)\)✍ Here is an alternative to Euler’s method:

\[\begin{split}\begin{split} v_{i+1} &= u_i + h f(t_i,u_i),\\ u_{i+1} &= u_i + hf(t_{i}+h,v_{i+1}). \end{split}\end{split}\]**(a)**Write out the method explicitly in the general one-step form (6.2.5) (i.e., clarify what \(\phi\) is for this method).**(b)**Show that the method is consistent.✍ Consider the problem \(u'=ku\), \(u(0)=1\) for constant \(k\) and \(t>0\).

**(a)**Find an explicit formula in terms of \(h\), \(k\), and \(i\) for the Euler solution \(u_i\) at \(t=ih\).**(b)**Find values of \(k\) and \(h\) such that \(|u_i|\to\infty\) as \(i\to\infty\) while the exact solution \(\hat{u}(t)\) is bounded as \(t\to\infty\).✍ Prove the fact, used in the proof of Theorem 6.2.7, that \(1+x\le e^x\) for all \(x\ge 0\).

✍ Suppose that the error in making a step is also subject to roundoff error \(\epsilon_{i+1}\), so that the total local error per unit step is \(Ch^p+\epsilon_{i+1} h^{-1}\); assume that \(|\epsilon_{i+1}| \le \epsilon\) for all \(i\) and that the initial condition is known exactly. Generalize Theorem 6.2.7 for this case.

- 1
Another point of view is that we can of course make local errors smaller by chopping \(h\) in half, but then we have to take twice as many steps. The important quantity, then, is local error

*per unit step length*, which is how \(\tau\) is defined.