2.3 Simulation Center
39
CVODES uses one of two numerical methods: using the difference approximation
back method of the inverse differentiation (BDF) [13] and the implicit Adams–
Moultonc method [11, 12]. Both methods can be represented by a linear multi-step
formula.
K 1
i=0
α n,i y n−i + t n
K 2
i=0
β n,i y
n−i = 0,
where y
n = f (t n , y n ). The coefficients α n,i and β n,i uniquely determine the specific
integration formula, with α n,0 = −1.
Adams–Fulton method is used for non-rigid tasks. He is answered with K 1 = 1
and K 2 = q −1. The order of the formula q can vary from 1 to 12. For hard problems,
BDF is applied with K 1 = q and K 2 = 0. The order of the formula q can vary from
1 to 5, and it is selected automatically and dynamically.
In both cases, a system of nonlinear equations is solved for each value t n
G(y n ) ≡ y n − t n β n,0 f (t n , y n ) − a n = 0,
a n =
i>0
α n,i y n−i + t n β n,i y
n−i
.
For non-rigid tasks, the iterative process used does not require solving systems
of linear equations. For hard problems, the Newton method is used, which requires
solving at each step a system of linear equations of the form
M
y
m+1
n
− y
m
n
= −G(y
m
n )
where y
m
n is the approximation obtained at step m and M is the approximation of the
Jacobi matrix:
M ≈ I − γ J, J = ∂ f /∂ y, γ = t n β n,0
If the right side and/or the initial conditions of the problem depend on the
parameter p ∈ R
M :
y
= f (t, y, p), y(t 0 ) = y 0 ( p),
CVODES can also perform forward or reverse (adjoint) sensitivity analysis.
A more detailed description of the software is presented in [21, 22].
Explicit Euler Method
The Euler method is the simplest method [11, 12] of solving the Cauchy problem for
first-order differential equations given explicitly:
y
= f (t, y), y(t 0 ) = y 0 .
39
CVODES uses one of two numerical methods: using the difference approximation
back method of the inverse differentiation (BDF) [13] and the implicit Adams–
Moultonc method [11, 12]. Both methods can be represented by a linear multi-step
formula.
K 1
i=0
α n,i y n−i + t n
K 2
i=0
β n,i y
n−i = 0,
where y
n = f (t n , y n ). The coefficients α n,i and β n,i uniquely determine the specific
integration formula, with α n,0 = −1.
Adams–Fulton method is used for non-rigid tasks. He is answered with K 1 = 1
and K 2 = q −1. The order of the formula q can vary from 1 to 12. For hard problems,
BDF is applied with K 1 = q and K 2 = 0. The order of the formula q can vary from
1 to 5, and it is selected automatically and dynamically.
In both cases, a system of nonlinear equations is solved for each value t n
G(y n ) ≡ y n − t n β n,0 f (t n , y n ) − a n = 0,
a n =
i>0
α n,i y n−i + t n β n,i y
n−i
.
For non-rigid tasks, the iterative process used does not require solving systems
of linear equations. For hard problems, the Newton method is used, which requires
solving at each step a system of linear equations of the form
M
y
m+1
n
− y
m
n
= −G(y
m
n )
where y
m
n is the approximation obtained at step m and M is the approximation of the
Jacobi matrix:
M ≈ I − γ J, J = ∂ f /∂ y, γ = t n β n,0
If the right side and/or the initial conditions of the problem depend on the
parameter p ∈ R
M :
y
= f (t, y, p), y(t 0 ) = y 0 ( p),
CVODES can also perform forward or reverse (adjoint) sensitivity analysis.
A more detailed description of the software is presented in [21, 22].
Explicit Euler Method
The Euler method is the simplest method [11, 12] of solving the Cauchy problem for
first-order differential equations given explicitly:
y
= f (t, y), y(t 0 ) = y 0 .