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Numerical Methods for Differential Equations
Chapter 1: Initial value problems in ODEs
Gustaf Soderlind and Carmen Arevalo
Numerical Analysis, Lund University
Textbooks: A First Course in the Numerical Analysis of Differential Equations, by Arieh Iserles
and Introduction to Mathematical Modelling with Differential Equations, by Lennart Edsberg
c© Gustaf Soderlind, Numerical Analysis, Mathematical Sciences, Lund University, 2008-09
Numerical Methods for Differential Equations – p. 1/52
Chapter 1: contents
◮ Course contents
◮ Introduction to initial value problems
◮ The explicit Euler method
◮ Convergence
◮ Order of consistency
◮ The trapezoidal rule
◮ Theta methods
◮ Numerical tests
◮ The linear test equation and numerical stability
◮ Stiff equations
Numerical Methods for Differential Equations – p. 2/52
What is Numerical Analysis?
Categories of mathematical problems
Category Algebra Analysis
linear computable not computable
nonlinear not computable not computable
Algebra: Only finite constructions
Analysis: Limits, derivatives, integrals etc. (transfinite)
Computable: the exact solution can be obtained in afinite number of operations
Numerical Methods for Differential Equations – p. 3/52
What is Numerical Analysis?
Category Algebra Analysis
linear computable not computable
nonlinear not computable not computable
With numerical methods, problems from all four categoriescan be solved:
“Numerical analysis aims to construct and analyzequantitative methods for the automatic computation ofapproximate solutions to mathematical problems.”
Goal: Construction of mathematical software
Numerical Methods for Differential Equations – p. 4/52
0. What will we study in this course?
To solve a differential equation analytically we look for adifferentiable function that satisfies the equation
Large, complex and nonlinear systems cannot be solvedanalytically
Numerical Methods for Differential Equations – p. 5/52
0. What will we study in this course?
To solve a differential equation analytically we look for adifferentiable function that satisfies the equation
Large, complex and nonlinear systems cannot be solvedanalytically
Instead, we compute numerical solutions with standardmethods and software
To solve a differential equation numerically we generate asequence {yk}N
k=0 of pointwise approximations to theanalytical solution: y(tk) ≈ yk
Numerical Methods for Differential Equations – p. 5/52
Some differential equations we will solve
◮ Initial value problems (IVP) first-order equations;higher-order equations; systems of differentialequations
Numerical Methods for Differential Equations – p. 6/52
Some differential equations we will solve
◮ Initial value problems (IVP) first-order equations;higher-order equations; systems of differentialequations
◮ Boundary value problems (BVP) two-point boundaryvalue problems; Sturm-Liouville eigenvalue problems
Numerical Methods for Differential Equations – p. 6/52
Some differential equations we will solve
◮ Initial value problems (IVP) first-order equations;higher-order equations; systems of differentialequations
◮ Boundary value problems (BVP) two-point boundaryvalue problems; Sturm-Liouville eigenvalue problems
◮ Partial differential equations (PDE) the diffusionequation; the advection equation; the wave equation
Numerical Methods for Differential Equations – p. 6/52
Some differential equations we will solve
◮ Initial value problems (IVP) first-order equations;higher-order equations; systems of differentialequations
◮ Boundary value problems (BVP) two-point boundaryvalue problems; Sturm-Liouville eigenvalue problems
◮ Partial differential equations (PDE) the diffusionequation; the advection equation; the wave equation
◮ Applications in all three areas
Numerical Methods for Differential Equations – p. 6/52
Initial value problems: examples
A first-order equation: a simple equation without a knownanalytical solution
dy
dt= y − e−t2 , y(0) = y0
Numerical Methods for Differential Equations – p. 7/52
Initial value problems: examples
A first-order equation: a simple equation without a knownanalytical solution
A second-order equation: motion of a pendulum
θ′′(t) +g
Lsin θ(t) = 0, θ(0) = θ0, θ′(0) = θ′0
where θ(t) is the angle, g is the gravitational constant andL is the pendulum length
Numerical Methods for Differential Equations – p. 7/52
Initial value problems: examples
A first-order equation: a simple equation without a knownanalytical solution
A second-order equation: pendulum equation
A system of equations: the predator–prey model
y′1(t) = k1 y1(t) − k2 y1(t) y2(t)
y′2(t) = k3 y1(t) y2(t) − k4y2(t)
where y1(t) is the population of the prey species and y2(t)
is the population of the predator species at time t
Numerical Methods for Differential Equations – p. 7/52
Boundary value problems: examples
Second-order two-point BVP: the electrostatic potentialu(r) between two concentric metal spheres satisfies
d2u
dr2+
2
r
du
dr= 0, u(R1) = V1, u(R2) = 0
at the distance r from the center; R1 and R2 are the radii ofthe two spheres
Numerical Methods for Differential Equations – p. 8/52
Boundary value problems: examples
Second-order two-point BVP: the electrostatic potentialu(r) between two concentric metal spheres satisfies
d2u
dr2+
2
r
du
dr= 0, u(R1) = V1, u(R2) = 0
at the distance r from the center; R1 and R2 are the radii ofthe two spheres
A Sturm-Liouville eigenproblem: Euler buckling of column
y′′ + λy = 0, y′(0) = 0, y(1) = 0
Find eigenvalues λ and eigenfunctions y
Numerical Methods for Differential Equations – p. 8/52
Partial differential equations: examples
The heat equation
ut(x, t) = uxx(x, t), x ∈ [0, a), t ∈ (0, b)
u(x, 0) = f(x), x ∈ [0, a]
u(0, t) = c1, u(a, t) = c2, t ∈ [0, b]
is a parabolic PDE modelling e.g. the temperature in aninsulated rod with constant temperatures c1 and c2 at itsends, and initial temperature distribution f(x)
Numerical Methods for Differential Equations – p. 9/52
Partial differential equations: examples
The heat equation
The wave equation
utt(x, t) = uxx(x, t), x ∈ (0, a), t ∈ (0, b)
u(0, t) = 0, u(a, t) = 0 for t ∈ [0, b]
u(x, 0) = f(x) for x ∈ [0, a]
ut(x, 0) = g(x) for x ∈ (0, a)
is a hyperbolic PDE e.g. modelling the displacement u of avibrating elastic string fixed at x = 0 and x = a
Numerical Methods for Differential Equations – p. 9/52
Some applications in ODEs
Initial value problems
• mechanics Mq = F (q)
• electrical circuits Cv = −I(v)• chemical reactions c = f(c)
Boundary value problems
• materials u′′ = M(u)/EI
• microphysics − ~
2mψ′′ = E
• eigenmodes u′′ + λu = 0
Numerical Methods for Differential Equations – p. 10/52
1. Initial value problems
Standard formulation
y′ = f(t, y) ; y(0) = y0
Scalar equation: y ∈ R; f scalar;Systems of ODEs: y ∈ R
m; f vector-valued.
Theorem (Cauchy & Peano) A solution to y′ = f(t, y)
exists if f is continuous.
Numerical Methods for Differential Equations – p. 11/52
Existence
Note: Continuity is not sufficient to guarantee uniqueness!
Example:
y = 2√y; y(0) = 0 with solution
y(t) = 0, t ≤ τ and y(t) = (t− τ)2, t > τ
This may happen in real physical models!
Consider the energy of a falling particle of mass m:12my2 +mgy = E := 0
This yields y = −√2g
√−y with y(0) = 0
Numerical Methods for Differential Equations – p. 12/52
Existence and uniqueness
Theorem If f(t, y) is continuous on [0, T ] and satisfies theLipschitz condition
‖f(t, u) − f(t, v)‖ ≤ L[f ] · ‖u− v‖
for all u, v and some Lipschitz constant L[f ] <∞, thereexists a unique solution to the initial value problem
y′ = f(t, y)
on [0, T ] for every initial value y(0) = y0
Numerical Methods for Differential Equations – p. 13/52
Existence and uniqueness . . .
Note Most problems do not satisfy a Lipschitz conditionon all of R
m
Example The problem
y = y2; y(0) = y0 > 0
has solution
y(t) = y0
1−y0t
The solution blows up at t = 1/y0 (“Finite escape time”)
Numerical Methods for Differential Equations – p. 14/52
Existence and uniqueness . . .
Other problems always satisfy Lipschitz conditions on Rm
Example
y = Ay; y(0) = y0
Lipschitz constant:
L[f ] = maxu 6=v‖Au−Av‖‖u−v‖
= maxy 6=0‖Ay‖‖y‖
= ‖A‖
The matrix norm ‖A‖ is a Lipschitz constant for f(y) = Ay
Numerical Methods for Differential Equations – p. 15/52
Standard form: x′ = F (t, x)
Example
y′′ = f(t, y, y′); y(0) = y0; y′(0) = y′0
Standard substitution Introduce new variables
x1 = y
x2 = y′
Then we get a system of first order equations
x′1 = x2
x′2 = f(t, x1, x2)
with x1(0) = y0 and x2(0) = y′0
Numerical Methods for Differential Equations – p. 16/52
2. The Explicit Euler method (1768)
y′ = f(t, y); y(t0) = y0
Replace derivative by finite difference approximation
y′(tn) ≈ y(tn + h) − y(tn)
h
Let {un} denote the numerical approximation to {y(tn)} and
h = tn+1 − tn denote the time-step.
Numerical Methods for Differential Equations – p. 17/52
2. The Explicit Euler method (1768)
y′ = f(t, y); y(t0) = y0
Replace derivative by finite difference approximation
y′(tn) ≈ y(tn + h) − y(tn)
h
Let {un} denote the numerical approximation to {y(tn)} and
h = tn+1 − tn denote the time-step. Compute un from
un+1 − un
h= f(tn, un), u0 = y0
Numerical Methods for Differential Equations – p. 17/52
2. The Explicit Euler method (1768)
y′ = f(t, y); y(t0) = y0
Replace derivative by finite difference approximation
y′(tn) ≈ y(tn + h) − y(tn)
h
Let {un} denote the numerical approximation to {y(tn)} and
h = tn+1 − tn denote the time-step. Compute un from
un+1 − un
h= f(tn, un), u0 = y0
The Explicit Euler method Given u0, t0 and h, compute
un+1 = un + hf(tn, un)
tn+1 = tn + h
Numerical Methods for Differential Equations – p. 17/52
Explicit Euler: Taylor series expansion
y′(t) = f(t, y), y(t0) = y0
Use Taylor series expansion
y(t + h) = y(t) + hy′(t) +h2
2!y′′(ζ)
= y(t) + hf(t, y(t)) + O(h2) ⇒y(t + h) ≈ y(t) + hf(t, y(t))
Numerical Methods for Differential Equations – p. 18/52
Explicit Euler: Taylor series expansion
y′(t) = f(t, y), y(t0) = y0
Use Taylor series expansion
y(t + h) = y(t) + hy′(t) +h2
2!y′′(ζ)
= y(t) + hf(t, y(t)) + O(h2) ⇒y(t + h) ≈ y(t) + hf(t, y(t))
Construct the numerical method (drop higher order terms)
un+1 = un + hf(tn, un); u0 = y0
tn+1 = tn + h
Numerical Methods for Differential Equations – p. 18/52
Explicit Euler: Taylor series expansion
y′(t) = f(t, y), y(t0) = y0
Use Taylor series expansion
y(t + h) = y(t) + hy′(t) +h2
2!y′′(ζ)
= y(t) + hf(t, y(t)) + O(h2) ⇒y(t + h) ≈ y(t) + hf(t, y(t))
Construct the numerical method (drop higher order terms)
un+1 = un + hf(tn, un); u0 = y0
tn+1 = tn + h
Explicit Euler!
Numerical Methods for Differential Equations – p. 18/52
Explicit Euler: graphic interpretation
“Take a step of size h in the direction of the tangent”
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y
Each step introduces an error and ends up on a differentsolution trajectory (dashed curves)
Numerical Methods for Differential Equations – p. 19/52
Explicit Euler: an example
Compute numerical solution to y′ = −y cos t; y(0) = 1; t ∈ [0, 2π]
Solution Choose a stepsize h = 2π/N ; N = 24 ⇒ h = π/12
Recursion un+1 = un + h (−un cos tn); u0 = 1
tn+1 = tn + h; t0 = 0
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The numerical solution is a sequence of points (tn, un)
Numerical Methods for Differential Equations – p. 20/52
3. Convergence
Analytical and numerical solutions for h = π/8 and π/128
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N=64*16
As h → 0 the numerical solution approaches the exact solution
Numerical Methods for Differential Equations – p. 21/52
3. Convergence
Analytical and numerical solutions for h = π/8 and π/128
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N=64
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N=64*16
As h → 0 the numerical solution approaches the exact solution
A method is convergent if, for every ODE with a Lipschitz
function f and every fixed T , with T = N · h, it holds that
limN→∞
‖yN,h − y(T )‖ = 0
Numerical Methods for Differential Equations – p. 21/52
Local and global errors
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1
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yyn+1•
yn•yn+1
•
y(t)y(tn)• •
y(tn+1)
Global error en = yn − y(tn) and en+1 = yn+1 − y(tn+1)
Local error ln+1 = yn+1 − y(tn+1)
Numerical Methods for Differential Equations – p. 22/52
The local error of the explicit Euler method
Insert exact data
yn+1 = y(tn) + hf(tn, y(tn))
Local error definition ln+1 = yn+1 − y(tn+1) implies residual
y(tn+1) = y(tn) + hf(tn, y(tn))− ln+1
Taylor series y(tn+1) = y(tn) + hy′(tn) + h2
2y′′(tn) + . . . ⇒
Local error for explicit Euler ln+1 ≈ − h2
2y′′(tn)
(along exact solution, solid curve)
Numerical Methods for Differential Equations – p. 23/52
Error propagation
Explicit Euler (numerical solution)
yn+1 = yn + hf(tn, yn)
Subtract Taylor series expansion of exact solution
y(tn+1) = y(tn) + hf(tn, y(tn)) +h2
2y′′(tn) + . . .
Global error recursion
en+1 = en + hf(tn, y(tn) + en) − hf(tn, y(tn)) + ln+1
Numerical Methods for Differential Equations – p. 24/52
Error propagation . . .
en+1 = en + hf(tn, y(tn) + en) − hf(tn, y(tn)) + ln+1
Take norms and use Lipschitz condition:
‖en+1‖ ≤ ‖en‖ + hL[f ] · ‖en‖ + ‖ln+1‖
Lemma If {an}, a0 = 0, is a sequence of non-negativenumbers satisfying
an+1 ≤ (1 + hµ)an + ch2 for µ > 0, then
an ≤ c
µh [(1 + hµ)n − 1], n = 0, 1, . . .
Numerical Methods for Differential Equations – p. 25/52
Convergence of Euler’s method: Theorem
Theorem The explicit Euler method is convergent
Numerical Methods for Differential Equations – p. 26/52
Convergence of Euler’s method: Theorem
Theorem The explicit Euler method is convergent
Proof Suppose f sufficiently differentiable. Given h and afixed T = Nh, let en,h = yn,h − y(tn)
Numerical Methods for Differential Equations – p. 26/52
Convergence of Euler’s method: Theorem
Theorem The explicit Euler method is convergent
Proof Suppose f sufficiently differentiable. Given h and afixed T = Nh, let en,h = yn,h − y(tn)
Apply the lemma to global error recursion, to get
‖en,h‖ ≤ c
L[f ]h [(1 + hL[f ])n − 1], n = 0, 1, . . .
withc = max
n‖ln‖/h2 ≈ max
t‖y′′‖/2
Numerical Methods for Differential Equations – p. 26/52
Convergence of Euler’s method . . .
As (1 + hL[f ])n < enhL[f ] ≤ eTL[f ], we have for n ≤ N ,
‖en,h‖ ≤ c
L[f ]h (eTL[f ] − 1)
So‖en,h‖ ≤ C(T ) · h
implies convergence because
limh→0
‖en,h‖ = 0
Note 1) The global error can be made arbitrarily small!2) The error is way too large for practical purposes!3) Better bounds can be obtained
Numerical Methods for Differential Equations – p. 27/52
Theoretical error “bound”
Example
y′ = −100y, y(0) = 1.
Then L[f ] = 100 and the exact solution is y(t) = e−100t with
y′′(t) = 1002e−100t, so c = 1002/2, with bound
‖en,h‖ ≤ 1002
2 · 100 h (e100T − 1)
Numerical Methods for Differential Equations – p. 28/52
Theoretical error “bound”
Example
y′ = −100y, y(0) = 1.
Then L[f ] = 100 and the exact solution is y(t) = e−100t with
y′′(t) = 1002e−100t, so c = 1002/2, with bound
‖en,h‖ ≤ 1002
2 · 100 h (e100T − 1)
Error estimate at T = 1 is ‖en,h‖ ≤ 50h e100 ≈ 1.4 · 1045h !
Numerical Methods for Differential Equations – p. 28/52
Theoretical error “bound”
Example
y′ = −100y, y(0) = 1.
Then L[f ] = 100 and the exact solution is y(t) = e−100t with
y′′(t) = 1002e−100t, so c = 1002/2, with bound
‖en,h‖ ≤ 1002
2 · 100 h (e100T − 1)
Error estimate at T = 1 is ‖en,h‖ ≤ 50h e100 ≈ 1.4 · 1045h !
But yn = (1 − 100h)n, so for h < 1/50, at T = 1 ,
Actual error is ‖en,h‖ = |(1− 100/N)N − e−100| ≤ 3.7 · 10−44h !
Numerical Methods for Differential Equations – p. 28/52
Computational test y = λ(y − sin t) + cos t
λ = −0.2 , with initial condition y(π/4) = 1/√
2
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h = π/10 h = π/20
Note Local error O(h2), global error O(h)!
Numerical Methods for Differential Equations – p. 29/52
4. Order of consistency
Given a method in the form yn+1 = Φn(f, h, y0, y1, . . . , yn)
Insert exact data y(t0), y(t1), . . . , y(tn)
Definition The order of consistency is p if
y(tn+1) − Φn(f, h, y(t0), y(t1), . . . , y(tn)) = O(hp+1), ∀n
as h→ 0, for every analytic f
Theorem The local error is then O(hp+1)
Numerical Methods for Differential Equations – p. 30/52
4. Order of consistency
Given a method in the form yn+1 = Φn(f, h, y0, y1, . . . , yn)
Insert exact data y(t0), y(t1), . . . , y(tn)
Definition The order of consistency is p if
y(tn+1) − Φn(f, h, y(t0), y(t1), . . . , y(tn)) = O(hp+1), ∀n
as h→ 0, for every analytic f
Theorem The local error is then O(hp+1)
Alternative The order of consistency is p if the formula isexact for all polynomials y = P (t) of degree p or less
Numerical Methods for Differential Equations – p. 30/52
Order of consistency of Euler’s method
Example Explicit Euler Φn(f, h, y0, . . . , yn) = yn + h f(tn, yn)
Numerical Methods for Differential Equations – p. 31/52
Order of consistency of Euler’s method
Example Explicit Euler Φn(f, h, y0, . . . , yn) = yn + h f(tn, yn)
Expanding in Taylor series,
y(tn+1) − [y(tn) + h f(tn, y(tn))] = O(h2)
so the method’s consistency order is one
Numerical Methods for Differential Equations – p. 31/52
Order of consistency of Euler’s method
Example Explicit Euler Φn(f, h, y0, . . . , yn) = yn + h f(tn, yn)
Expanding in Taylor series,
y(tn+1) − [y(tn) + h f(tn, y(tn))] = O(h2)
so the method’s consistency order is one
◮ Alternatively, suppose y(t) = 1 with f = y′ = 0. Then
1 = y(tn+1) = y(tn) + hf(tn, y(tn)) = 1 + h · 0 = 1 ⇒ exact!
Numerical Methods for Differential Equations – p. 31/52
Order of consistency of Euler’s method
Example Explicit Euler Φn(f, h, y0, . . . , yn) = yn + h f(tn, yn)
Expanding in Taylor series,
y(tn+1) − [y(tn) + h f(tn, y(tn))] = O(h2)
so the method’s consistency order is one
◮ Alternatively, suppose y(t) = 1 with f = y′ = 0. Then
1 = y(tn+1) = y(tn) + hf(tn, y(tn)) = 1 + h · 0 = 1 ⇒ exact!
◮ Next take 1st degree polynomial y(t) = t with f = y′ = 1, then
h = y(tn+1) = y(tn) + hf(tn, y(tn)) = 0 + h · 1 = h ⇒ exact!
Numerical Methods for Differential Equations – p. 31/52
Order of consistency of Euler’s method
Example Explicit Euler Φn(f, h, y0, . . . , yn) = yn + h f(tn, yn)
Expanding in Taylor series,
y(tn+1) − [y(tn) + h f(tn, y(tn))] = O(h2)
so the method’s consistency order is one
◮ Alternatively, suppose y(t) = 1 with f = y′ = 0. Then
1 = y(tn+1) = y(tn) + hf(tn, y(tn)) = 1 + h · 0 = 1 ⇒ exact!
◮ Next take 1st degree polynomial y(t) = t with f = y′ = 1, then
h = y(tn+1) = y(tn) + hf(tn, y(tn)) = 0 + h · 1 = h ⇒ exact!
◮ For a 2nd degree polynomial, y(t) = t2 with f = y′ = 2t. Then
h2 = y(tn+1) = y(tn)+hf(tn, y(tn)) = 0+h ·0 = 0 6= h2 ⇒ p = 1 !
Numerical Methods for Differential Equations – p. 31/52
5. The trapezoidal rule
Explicit Euler linearizes y at tn with a slope of y′(tn)
Numerical Methods for Differential Equations – p. 32/52
5. The trapezoidal rule
Explicit Euler linearizes y at tn with a slope of y′(tn)
Instead, linearize with a slope equal to the average of y′(tn)
and y′(tn+1)
y(t) ≈ y(tn) + (t− tn)1
2[f(tn, y(tn)) + f(tn+1, y(tn+1))]
Numerical Methods for Differential Equations – p. 32/52
5. The trapezoidal rule
Explicit Euler linearizes y at tn with a slope of y′(tn)
Instead, linearize with a slope equal to the average of y′(tn)
and y′(tn+1)
y(t) ≈ y(tn) + (t− tn)1
2[f(tn, y(tn)) + f(tn+1, y(tn+1))]
The numerical method is the trapezoidal rule
tn+1 = tn + h
yn+1 = yn +h
2[f(tn, yn) + f(tn+1, yn+1)]
Numerical Methods for Differential Equations – p. 32/52
5. The trapezoidal rule
Explicit Euler linearizes y at tn with a slope of y′(tn)
Instead, linearize with a slope equal to the average of y′(tn)
and y′(tn+1)
y(t) ≈ y(tn) + (t− tn)1
2[f(tn, y(tn)) + f(tn+1, y(tn+1))]
The numerical method is the trapezoidal rule
tn+1 = tn + h
yn+1 = yn +h
2[f(tn, yn) + f(tn+1, yn+1)]
The method is implicit . Nonlinear equation solving required
Numerical Methods for Differential Equations – p. 32/52
Trapezoidal rule: Order and convergence
Order of consistency (sketch):
y(tn+1) − {y(tn) +h
2[f(tn, y(tn)) + f(tn+1, y(tn+1))]}
= y(tn) + h y′(tn) +h2
2y′′(tn) + O(h3)
−{y(tn) +h
2(y′(tn) + [y′(tn) + h y′′(tn) + O(h2)])}
= O(h3)
Numerical Methods for Differential Equations – p. 33/52
Trapezoidal rule: Order and convergence
Order of consistency (sketch):
y(tn+1) − {y(tn) +h
2[f(tn, y(tn)) + f(tn+1, y(tn+1))]}
= y(tn) + h y′(tn) +h2
2y′′(tn) + O(h3)
−{y(tn) +h
2(y′(tn) + [y′(tn) + h y′′(tn) + O(h2)])}
= O(h3)
The method is of order two
Numerical Methods for Differential Equations – p. 33/52
Trapezoidal rule: Order and convergence
Order of consistency (sketch):
y(tn+1) − {y(tn) +h
2[f(tn, y(tn)) + f(tn+1, y(tn+1))]}
= y(tn) + h y′(tn) +h2
2y′′(tn) + O(h3)
−{y(tn) +h
2(y′(tn) + [y′(tn) + h y′′(tn) + O(h2)])}
= O(h3)
The method is of order two
Theorem The trapezoidal rule is convergent(No proof given here)
Numerical Methods for Differential Equations – p. 33/52
Trapezoidal rule: the dramatic effect of 2nd order
Compute numerical solution to y′ = −y cos t; y(0) = 1; t ∈ [0, 8π].
Choose h = π/12. Note that f is linear in y. The recursion is
un+1 =1 − h
2 cos tn
1 + h2 cos tn+1
un ; u0 = 1
0 5 10 15 20 250
0.5
1
1.5
2
2.5
3
N=96 Trapezoidal rule
0 5 10 15 20 250
0.5
1
1.5
2
2.5
3
N=96 Explicit Euler
Solutions with Trapezoidal rule and Explicit Euler
Numerical Methods for Differential Equations – p. 34/52
The implicit midpoint rule
We can also approximate the derivative y′(t) by taking the
average of tn and tn+1 as well as yn and yn+1:
y′(t) ≈ f(tn + tn+1
2,yn + yn+1
2), t ∈ [tn, tn+1]
Numerical Methods for Differential Equations – p. 35/52
The implicit midpoint rule
We can also approximate the derivative y′(t) by taking the
average of tn and tn+1 as well as yn and yn+1:
y′(t) ≈ f(tn + tn+1
2,yn + yn+1
2), t ∈ [tn, tn+1]
The resulting method is the 2nd order implicit midpoint method
tn+1 = tn + h
yn+1 = yn + h f(tn + tn+1
2,yn + yn+1
2)
Numerical Methods for Differential Equations – p. 35/52
6. Theta methods
We construct methods that linearize y at tn with a slope equal to
a convex combination of y′(tn) and y′(tn+1):
yn+1 = yn + h [θf(tn, yn) + (1 − θ)f(tn+1, yn+1)], θ ∈ [0, 1]
◮ Explicit Euler: θ = 1
◮ Trapezoidal rule (implicit): θ = 1/2
◮ Implicit Euler: θ = 0 ⇒ yn+1 = yn + h f(tn+1, yn+1)
Numerical Methods for Differential Equations – p. 36/52
Theta methods: Order and convergence
Use Taylor expansion to get
y(tn+1) − y(tn) − h [θf(tn, y(tn)) + (1 − θ)f(tn+1, y(tn+1))]
= (θ − 1
2)h2y′′(tn) +
1
2(θ − 2
3)h3y′′′(tn) + O(h4)
Numerical Methods for Differential Equations – p. 37/52
Theta methods: Order and convergence
Use Taylor expansion to get
y(tn+1) − y(tn) − h [θf(tn, y(tn)) + (1 − θ)f(tn+1, y(tn+1))]
= (θ − 1
2)h2y′′(tn) +
1
2(θ − 2
3)h3y′′′(tn) + O(h4)
If θ = 1/2 the method is of order 2; otherwise it is of order 1
Numerical Methods for Differential Equations – p. 37/52
Theta methods: Order and convergence
Use Taylor expansion to get
y(tn+1) − y(tn) − h [θf(tn, y(tn)) + (1 − θ)f(tn+1, y(tn+1))]
= (θ − 1
2)h2y′′(tn) +
1
2(θ − 2
3)h3y′′′(tn) + O(h4)
If θ = 1/2 the method is of order 2; otherwise it is of order 1
Theorem (without proof) The θ methods are convergent
Numerical Methods for Differential Equations – p. 37/52
Implicit methods
Implicit Euler yn+1 = yn + h f(tn+1, yn+1)
We need to solve a nonlinear equation to compute yn+1
The extra cost is motivated if we can take larger steps
There are some problems where implicit methods can takeenormous time steps without losing accuracy!
We will return to how to solve nonlinear equations
Numerical Methods for Differential Equations – p. 38/52
7. Tests: Explicit Euler y = λ(y − sin t) + cos t
λ = −0.2 , with initial condition y(π/4) = 1/√
2
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t
y
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t
y
h = π/10 h = π/20
Local error O(h2), global error O(h)!
Numerical Methods for Differential Equations – p. 39/52
Tests: Implicit Euler y = λ(y − sin t) + cos t
λ = −0.2 , with initial condition y(π/4) = 1/√
2
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t
y
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t
y
h = π/10 h = π/20
Local error O(h2), global error O(h)!
Numerical Methods for Differential Equations – p. 40/52
Tests: Implicit Euler y = λ(y − sin t) + cos t
λ = −10 , with initial condition y(π/4) = 1/√
2
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t
y
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t
y
h = π/10 h = π/20
Local error = O(h2), global error = O(h), but difficult to see!
Numerical Methods for Differential Equations – p. 41/52
Tests: Explicit Euler y = λ(y − sin t) + cos t
λ = −10 , with initial condition y(π/4) = 1/√
2
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t
y
h = π/20
Numerical Methods for Differential Equations – p. 42/52
Tests: Explicit Euler y = λ(y − sin t) + cos t
λ = −10 , with initial condition y(π/4) = 1/√
2
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t
y
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t
y
h = π/10 h = π/20
NUMERICAL INSTABILITY! Stability for h small
Numerical Methods for Differential Equations – p. 42/52
8. The linear test equation
Defintion The linear test equation is
y′ = λy; y(0) = 1, t ≥ 0, where λ ∈ C
Numerical Methods for Differential Equations – p. 43/52
8. The linear test equation
Defintion The linear test equation is
y′ = λy; y(0) = 1, t ≥ 0, where λ ∈ C
As y(t) = eλt, we have
|y(t)| ≤ K ⇔ Re(λ) ≤ 0
Mathematical stability Bounded solutions if Re(λ) ≤ 0
Numerical Methods for Differential Equations – p. 43/52
8. The linear test equation
Defintion The linear test equation is
y′ = λy; y(0) = 1, t ≥ 0, where λ ∈ C
As y(t) = eλt, we have
|y(t)| ≤ K ⇔ Re(λ) ≤ 0
Mathematical stability Bounded solutions if Re(λ) ≤ 0
When does a numerical method have the same property?
Does Re(λ) ≤ 0 imply numerical stability ?
Numerical Methods for Differential Equations – p. 43/52
Numerical stability: The stability region
Definition The stability region D of a method is the set of all
hλ ∈ C such that |yn| ≤ K when the method is applied to the test
equation
4
Numerical Methods for Differential Equations – p. 44/52
Numerical stability: The stability region
Definition The stability region D of a method is the set of all
hλ ∈ C such that |yn| ≤ K when the method is applied to the test
equation
Example For Euler’s method, yn+1 = (1 + hλ)yn, so yn remains
bounded if and only if |1 + hλ| ≤ 1
4
Numerical Methods for Differential Equations – p. 44/52
Numerical stability: The stability region
Definition The stability region D of a method is the set of all
hλ ∈ C such that |yn| ≤ K when the method is applied to the test
equation
Example For Euler’s method, yn+1 = (1 + hλ)yn, so yn remains
bounded if and only if |1 + hλ| ≤ 1
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
DEuler = {z ∈ C : |1 + z| ≤ 1}
Numerical Methods for Differential Equations – p. 44/52
A-stability
Definition A method is called A-stable if its stability region
contains C−, i.e. C− ≡ {z ∈ C : Re(z) ≤ 0} ⊂ D.
Numerical Methods for Differential Equations – p. 45/52
A-stability
Definition A method is called A-stable if its stability region
contains C−, i.e. C− ≡ {z ∈ C : Re(z) ≤ 0} ⊂ D.
For the trapezoidal rule
DTR =
{
z ∈ C :
∣
∣
∣
∣
∣
1 + 12z
1 − 12z
∣
∣
∣
∣
∣
≤ 1
}
≡ C−
so if Re(z) ≤ 0 then yn remains bounded for any h > 0
Numerical Methods for Differential Equations – p. 45/52
A-stability
Definition A method is called A-stable if its stability region
contains C−, i.e. C− ≡ {z ∈ C : Re(z) ≤ 0} ⊂ D.
For the trapezoidal rule
DTR =
{
z ∈ C :
∣
∣
∣
∣
∣
1 + 12z
1 − 12z
∣
∣
∣
∣
∣
≤ 1
}
≡ C−
so if Re(z) ≤ 0 then yn remains bounded for any h > 0
The explicit Euler method is not A-stable but the implicit Euler
method and the trapezoidal rule are A-stable
Usually: “If the original problem is stable, then an A-stable
method will replicate that behavior numerically”
Numerical Methods for Differential Equations – p. 45/52
Relevance of the linear test equation
When we have a system of equations
y′ = Ay
and A has a full set of eigenvectors with eigenvalues
Λ = diag(λ1, λ2, . . . , λN ), then
◮ A can be diagonalized, T−1AT = Λ
◮ Putting y = Tx transforms the system to x′ = Λx (scalar
systems)
◮ . . . equal to x′k = λkxk for k = 1, . . . , d
◮ The exact solution y(t) = etAy0 is stable if and only if
Re(λk) ≤ 0 for all k = 1, . . . , d
Numerical Methods for Differential Equations – p. 46/52
Relevance of the linear test equation . . .
Applying a method to y′ = Ay gives (say)
yn+1 = (I + hA)yn
Numerical Methods for Differential Equations – p. 47/52
Relevance of the linear test equation . . .
Applying a method to y′ = Ay gives (say)
yn+1 = (I + hA)yn
Noting that T−1(I + hA)T = I + hΛ, putting yn = Txn produces
xn+1 = (I + hΛ)xn
Numerical Methods for Differential Equations – p. 47/52
Relevance of the linear test equation . . .
Applying a method to y′ = Ay gives (say)
yn+1 = (I + hA)yn
Noting that T−1(I + hA)T = I + hΛ, putting yn = Txn produces
xn+1 = (I + hΛ)xn
The same as applying the method to the diagonalized system
x′ = Λx
Numerical Methods for Differential Equations – p. 47/52
Relevance of the linear test equation . . .
Applying a method to y′ = Ay gives (say)
yn+1 = (I + hA)yn
Noting that T−1(I + hA)T = I + hΛ, putting yn = Txn produces
xn+1 = (I + hΛ)xn
The same as applying the method to the diagonalized system
x′ = Λx
Diagonalization and discretization commute! Stability condition
from linear test equation applies to all diagonalizable systems!
Numerical Methods for Differential Equations – p. 47/52
9. Stiff ODEs
Example Solve y = λ(y − sin t) + cos t with λ = −50
Solution
Particular: yP (t) = sin t
Homogeneous: yH(t) = eλt
General: y(t) = eλ(t−t0)(y(t0) − sin t0) + sin t
Study the flow of this equation and numerical solutions
Numerical Methods for Differential Equations – p. 48/52
Flow (solution trajectories)
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t
y
Numerical Methods for Differential Equations – p. 49/52
Stiffness. . .
With Explicit Euler the solution approaches sin t as t→ ∞,
as the exact solution does, only if 0 < h <1
25
Numerical Methods for Differential Equations – p. 50/52
Stiffness. . .
With Explicit Euler the solution approaches sin t as t→ ∞,
as the exact solution does, only if 0 < h <1
25
This means h must be kept small, not to keep errors small,but to have a stable numerical solution
Numerical Methods for Differential Equations – p. 50/52
Stiffness. . .
With Explicit Euler the solution approaches sin t as t→ ∞,
as the exact solution does, only if 0 < h <1
25
This means h must be kept small, not to keep errors small,but to have a stable numerical solution
The trapezoidal rule solution tends to sin t as t→ ∞, as theexact solution does, for every h > 0
This means h must be kept small only to keep errors small
Numerical Methods for Differential Equations – p. 50/52
Stiffness. . .
With Explicit Euler the solution approaches sin t as t→ ∞,
as the exact solution does, only if 0 < h <1
25
This means h must be kept small, not to keep errors small,but to have a stable numerical solution
The trapezoidal rule solution tends to sin t as t→ ∞, as theexact solution does, for every h > 0
This means h must be kept small only to keep errors small
For A-stable methods, there is no stability restriction on hThe step size is only restricted by accuracy
Numerical Methods for Differential Equations – p. 50/52
Stiffness. . .
Stiff differential equations are characterized byhomogeneous solutions being strongly damped
Example y = λ(y − sin t) + cos t with λ≪ −1
Stability regions of explicit methods are bounded, andhλ ∈ D puts a strong stability restriction on h
Example Explicit Euler applied to the problem aboverequires h ≤ −2/λ≪ 1
Numerical Methods for Differential Equations – p. 51/52
Stiffness. . .
Implicit methods with unbounded stability regions put norestrictions on h, and the stepsize is only restricted byaccuracy requirement!
Example The implicit Euler applied to the problem aboverequires only that
ln ≈ h2y′′
2≈ h2
2sin tn
is sufficiently small, independently of λ
Numerical Methods for Differential Equations – p. 52/52