NUMERICAL METHODS:
SOLVING THE SECONDORDER ODE BVP
Presentation by Addison Euhus, Nathan
Feldman, Xiaoyue Pi, Zachary Rom,
Guillaume Toujas
Overview of Second-Order ODE BVPs
Second Order ODE
F(y”, y’, y, x) = 0
Boundary Value Conditions
• Dirichlet:
y(a) = α
y(b) = β
• Neumann:
y’(a) = α
y’(b) = β
• Mixed:
y(a) = α
y’(b) = β
• Robin:
M*y(a) – N*y’(a) = α
M*y(b) + N*y’(b) = β
Real World Applications
Electrically Conducting Solids
y’’ = λeμy
y(0) = y(1) = 0
This arises in applications involving the diffusion of heat generated
by positive temperature-dependent sources
Circular Membrane Theory
y’’ + k/y2 + 3y’/x = 0, 0<x<1
At the edge we have y(1) = λ > 0, at the center we have y’(0) = 0
This is the equation for a circular membrane subject to normal
uniform pressure
Overview of Numerical Methods
We will cover four basic methods used to solve second-order
ODE BVPs in one-dimension space
• Shooting Method
Solving the IVP and tweaking the parameters to fit the BVP
• Finite Difference Method
Replacing derivatives with difference approximations and solving
the resulting linear system
• Finite Element Method
Covered in class, subdivides domain into finite elements and
minimizes associated error function
• Quadrature Method
Constructs an integral representation of the problem and applies
numerical integration techniques
Shooting Method
BVP Problem
Shooting: Core Idea
• Convert BVP into an IVP problem.
• This requires a guess y’(a) = θ
• Reduce high order ODE into system first order ODEs
• Solve the IVP (let y(b, θ) denote solution at be with our guess
• y(b, θ) == β then IVP solution is equivalent to BVP solution
• y(b, θ) != β then choose a new guess (pick smartly)
• Taking a guess and “firing at a target” is where the
method derives its name
Shooting Advantages and Disadvantages
• Advantages
• Solves both linear and non-linear BVPs
• Simple implementation
• Effect when the interval [a,b] is short since its “easier” to shoot a
target a short distance away
• Disadvantages
• Unstable for some problems. Typically those of highly nonlinear or
unstable ODEs.
• Requires a good initial guess for the IVP as the numerical method
does not guarantee convergence.
• The length of the interval of the boundary values has a high impact
on the results. A large interval requires a large number of iterations.
Reduction
Solve the IVP
• Numerical Scheme
• Euler, RK23, RK45, etc
• F(θ) represents the difference between IVP solution and β
• F(θ)= y(b; θ)- β
• If F contains a root with the value θ then the solution y(b;
θ) to our IVP problem is also a solution to the BVP
problem.
• Linear ODE use Linear Interpolation (proof in paper)
• Non-linear ODE use Secant, Newtons, or other root finding method
Linear ODE
• Guess y’(a) = 0 and y’(a) = 1 and reduce BVP to IVP
along with high order to system first order ODEs
• Linear combination of solutions of the ODE also satisfies
the ODE
• y(x) = y(x;0) + θy(x;1)
• y(b) = y(b;0) + θy(b;1) = β
• Result (proof in paper and its references)
Nonlinear Using Secant
Nonlinear Using Secant Procedure
• Convert the BVP into IVP form as a system of first order
•
•
•
•
ODEs.
Choose two guesses θ1 and θ2
Solve an IVP1 where y’(a)=θ1 and IVP2 where y’(a)=θ2
Compute F(θ1) and F(θ2)
While (we have not converged up to the tolerance we
want)
• Use the secant method to guess the next θ1 and θ2
• Solve the IVP using these two new guesses using the numerical
scheme of your choice (Euler, RK, etc)
• Compute Fθ1 and Fθ2
• Check for convergence of the root
Nonlinear Using Newton’s
Therefore we must know F’(θ) since we
are finding the root of F(θ)
Define z(x;θ) as
It satisfies ODE
Nonlinear Using Newton’s
By differentiating the original BVP with
respect to θ
Notice that we can obtain
Therefore in Newton’s method we can state F’(θ) in terms of z, and now we
can use Newton’s method
Another IVP!
• Newton’s method requires another IVP to be solved
during each iteration.
• We make up for this additional computation by taking
advantage of the fast convergence of Newton’s method
IVPs Nonlinear Newton’s
• First
• Second
Nonlinear Using Newton’s Procedure
• Choose a numerical scheme for solving systems of first
order ODEs, such as Euler or Runge-Kutta
• Convert BVP into an IVP1 problem and then convert the
BVP representing Newton’s method to the IVP2. See the
above conversions
• Until convergence do
• Solve both IVP1 and IVP2 using numerical scheme
• Use Newton’s Method to approximate the next guess for θ where
F'θ=z(b)
Example
• BVP
• Exact
Shooting With RK4-5 and Secant
Shooting with RK4-5 and Newton’s
Dealing With Short Interval Problem
• Multiple Shooting Method
• We only discussed the single shooting method, but for those who
are interested you can research the multiple-shooting method.
• Split interval into pieces and solve an IVP on each piece. See
references in paper
Multiple Shooting Method
Split the interval up
and solve the IVP in
each interval.
The Finite Difference Method
• Approximates the differential operator by replacing the
derivatives in the equation using difference quotients
• These difference quotients are derived from the Taylor
Series of u(x+h) and u(x-h)
Name
Value
Approximation
Error
Forward Difference
u’
(un+1 – un)/h
O(h)
Backward Difference
u’
(un – un-1)/h
O(h)
Central Difference
u’
(un+1 – un-1)/(2h)
O(h2)
Consistent SecondOrder
u’’
(un+1 – 2un + un-1)/h2
O(h2)
• Example: u” – 2u = x2 becomes
un+1/h2 – (2 – 2/h2)un + un-1/h2 = xi2
Finite Difference Procedure
1.
Discretize x
Dirichlet: Given u(a) and u(b)
Neumann: Given u(a) and u’(b)
2.
Discretize the ODE
Example: un+1/h2 – (2 – 2/h2)un + un-1/h2 = xi2
3.
Discretize the Boundary Conditions
For Neumann only:
4.
u½ ≅ (u0+u1)/2
u’N-½ ≅ (uN – uN-1)/h
Construct a Linear System and Solve
Characteristics of FD Method
PROS
CONS
Very easy to implement
Does not work very well on
large intervals
Results in solving a system
with a tridiagonal matrix
Accuracy is reduced due to
computation of linear
system
Existence, Consistency and Accuracy:
-u”(x) + c(x)u(x) = f(x) with Dirichlet Conditions
• If c(x) > 0 for x ∈ [x0,xN] then A is symmetric positive
definite, and therefore a solution exists
• If u ∈ C4(x0,xN) then using the error terms, it can be shown
that the finite difference numerical scheme is consistent
and second-order accurate
Example:
Exact Solution:
u(x) = x2sin(10x)
Second-Order ODE:
u” + 100u = 40xcos(10x) + 2sin(10x)
u(0) = 0,
u(1) = sin(10)
N=5
N = 500
N = 50
The Finite Element Method
• We have covered the procedure of the method in class
before the third midterm, so everyone already has some
familiarity with it
• The method can be used to solve boundary value
problem (BVP) by using Galerkin approximation
• The Galerkin method can also be used to solve Initial
Value Problem (IVP). Basically, we can achieve it by
either using weak form combined with stabilizing
techniques or using least-squares process
The Positive-Definite Constraint
• When the differential operator is self-adjoint (if and only if
A = A*) the Galerkin method with weak form and least
squares process are the only FEM that guarantee
positive-definite coefficient matrix
• When the differential operator is non-self adjoint or nonlinear the least square process is the only FEM that
guarantees a positive-definite coefficient matrix
• When considering an IVP, the differential operator is either
non-self adjoint or nonlinear. It is never self-adjoint. Hence,
the only FEM that can guarantee positive definite
coefficient matrices for all IVPs is least squares process
Practical Uses of the FEM
• Many people are under the impression that the Galerkin
method with weak form is the only FEM – this happens
because almost all commercial FEM software is based on this
method
• The reason for this is because the differential operators of
differential equations which describe elastic solid mechanics
are always self-adjoint, hence the Galerkin method with weak
form works very well
• Around the late 1990s, the additional resources needed for
least-squares were invented, and least-squares started being
used in the research area. However, since so many companies
were built on Galerkin method with weak form and they are
only marketed towards solid mechanics applications, they still
continue to use it
Differences Between FDM and FEM
• FEM uses calculations at each individual interval and
tends to be more computationally expensive
• FDM uses the current step to find the next one using finite
difference formulation
• For the most part FEM is a bit more accurate, but in
dynamic problems based on time, FDM can be preferred
for accuracy
The Quadrature Method: The Goal
• Find the integral form of the solution for a second-order
boundary-value problem. Use quadrature rules to solve
the equation for the boundary value problem
• Essentially, solve the ODE numerically by converting it
into an integral and solving the integral using already
established numerical methods
For a classic 2nd-order boundary
problem…
1. Have the following form for a second-order BVP:
𝑦 ′′ = 𝑓(𝑥, 𝑦, 𝑦′), 𝑓 𝑥𝑜 = 𝑦𝑜 , 𝑓 𝑥𝑀 = 𝑦𝑀 ; 𝑥𝑜 < 𝑥𝑀
2. Approximate the Value of 𝑦′ 𝑥𝑖 at strategically chosen 𝑥𝑖
For instance, use Runge-Kutta to approximate y ′ 𝑥𝑖 , x0 < xi < xM
for various 𝑥𝑖
3. Use numerical integration to approximate 𝑦 𝑥𝑗
For instance, given our values found for
𝑥𝑗
′
′
𝑦 𝑥𝑗 , 𝑦
, 𝑦′ 0 ,
2
use Simpson’s rule to approximate 𝑦(𝑥𝑗 )
A Basic Example
• Let’s say we are given the equation 𝑦 ′′ =
𝑓 𝑥, 𝑦, 𝑦 ′ ; y 0 = e, y 1 = 0
• We have used Runge-Kutta to approximate the value
y’(1/4) = 1/2
• Note: This isn’t the actual value obtained, but it is usable to
illustrate the concept
• We can then approximate y
1
2
=𝑦 0 +
0
𝑦′(𝑥)
1/2
=
1 ′ 1
𝑦
+ 𝑦 0 = 𝑒 + .25
2
4
• Note: This is just using the midpoint rule, an extremely poor method
of approximation
• We can use this technique to approximate any values of y
in (0,1)
Advantages and Disadvantages
Pros:
• This method allows us to use any method of numerical integration.
For instance, if we want to use Gaussian 2-point quadrature rules,
we can use that instead of the midpoint method (which is obviously
a vast improvement)
• The method allows us to approximate the solutions to difficult
differential equations, where the actual value of the integral may
be difficult to find
Cons:
• The different values of 𝑥𝑖 we find limit the method of numerical
integration that we can use. For instance, if we have a differential
equation over (0,1) and y’(0.00001), then using the midpoint
method only approximates y(0.00002). This means it becomes
more difficult to get accuracy near the endpoints of the interval
without extremely small step sizes, which take large amounts of
computational time
• Taking large step sizes isn’t easy either. If we want to use our
midpoint method again, we are stuck only using values of y’(x < ½)
Conclusion
• There are a variety of different methods that can be used
to determine numerical solutions to second-order ODE
BVPs. Each method has strengths and weaknesses, and
the selection of the method is greatly influenced by the
specific problem that is being analyzed.
• As we have seen throughout the presentation, many
established numerical procedures – such as solutions to
matrix equations and numerical integration techniques –
are incorporated into the analysis and computation of the
approximations to these ODEs.