Solve stiff differential equations; low order method
[T,Y] =
solver
(odefun,tspan,y0)
[T,Y] = solver
(odefun,tspan,y0,options)
[T,Y,TE,YE,IE] = solver
(odefun,tspan,y0,options)
sol = solver
(odefun,[t0
tf],y0...)
This page contains an overview of the solver functions: ode23
, ode45
, ode113
, ode15s
, ode23s
, ode23t
,
and ode23tb
. You can call any of these solvers
by substituting the placeholder, solver
,
with any of the function names.
The following table describes the input arguments to the solvers.
| A function handle that evaluates the right side of the
differential equations. All solvers solve systems of equations in
the form y′ = f(t,y)
or problems that involve a mass matrix, M(t,y)y′ = f(t,y).
The |
| A vector specifying the interval of integration, For |
Specifying Specifying | |
| A vector of initial conditions. |
| Structure of optional parameters that change the default integration properties. This is the fourth input argument. [t,y] = solver(odefun,tspan,y0,options) You can create options using the |
The following table lists the output arguments for the solvers.
| Column vector of time points. |
| Solution array. Each row in |
| The time at which an event occurs. |
| The solution at the time of the event. |
| The index |
| Structure to evaluate the solution. |
[T,Y] =
with solver
(odefun,tspan,y0)tspan = [t0 tf]
integrates the system of differential
equations y′ = f(t,y)
from time t0
to tf
with initial
conditions y0
. The first input argument, odefun
,
is a function handle. The function, f = odefun(t,y)
,
for a scalar t
and a column vector y
,
must return a column vector f
corresponding to f(t,y).
Each row in the solution array Y
corresponds to
a time returned in column vector T
. To obtain solutions
at the specific times t0
, t1,...,tf
(all
increasing or all decreasing), use tspan = [t0,t1,...,tf]
.
Parameterizing Functions explains how to provide additional
parameters to the function fun
, if necessary.
[T,Y] =
solves
as above with default integration parameters replaced by property values specified
in solver
(odefun,tspan,y0,options)options
, an argument created with the odeset
function.
Commonly used properties include a scalar relative error tolerance RelTol
(1e-3
by
default) and a vector of absolute error tolerances AbsTol
(all
components are 1e-6
by default). If certain components
of the solution must be nonnegative, use the odeset
function
to set the NonNegative
property to the indices
of these components. See odeset
for
details.
[T,Y,TE,YE,IE] =
solves
as above while also finding where functions of (t,y),
called event functions, are zero. For each event function, you specify
whether the integration is to terminate at a zero and whether the
direction of the zero crossing matters. Do this by setting the solver
(odefun,tspan,y0,options)'Events'
property
to a function, e.g., events
or @events
, and
creating
a function [value
,isterminal
,direction
]
= events
(t
,y
).
For the i
th event function in events
,
value(i)
is the value of the function.
isterminal(i) = 1
, if the integration
is to terminate at a zero of this event function and 0
otherwise.
direction(i) = 0
if all zeros are
to be computed (the default), +1
if only the zeros
where the event function increases, and -1
if only
the zeros where the event function decreases.
Corresponding entries in TE
, YE
,
and IE
return, respectively, the time at which
an event occurs, the solution at the time of the event, and the index i
of
the event function that vanishes.
sol =
returns a structure that you can use with solver
(odefun,[t0
tf],y0...)deval
to
evaluate the solution at any point on the interval [t0,tf]
.
You must pass odefun
as a function handle. The
structure sol
always includes these fields:
| Steps chosen by the solver. |
| Each column |
| Solver name. |
If you specify the Events
option and events
are detected, sol
also includes these fields:
| Points at which events, if any, occurred. |
| Solutions that correspond to events in |
| Indices into the vector returned by the function specified
in the |
If you specify an output function as the value of the OutputFcn
property,
the solver calls it with the computed solution after each time step.
Four output functions are provided: odeplot
, odephas2
, odephas3
, odeprint
.
When you call the solver with no output arguments, it calls the default odeplot
to
plot the solution as it is computed. odephas2
and odephas3
produce
two- and three-dimensional phase plane plots, respectively. odeprint
displays
the solution components on the screen. By default, the ODE solver
passes all components of the solution to the output function. You
can pass only specific components by providing a vector of indices
as the value of the OutputSel
property. For example,
if you call the solver with no output arguments and set the value
of OutputSel
to [1,3]
, the solver
plots solution components 1 and 3 as they are computed.
For the stiff solvers ode15s
, ode23s
, ode23t
,
and ode23tb
, the Jacobian matrix ∂f/∂y is
critical to reliability and efficiency. Use odeset
to
set Jacobian
to @FJAC
if FJAC(T,Y)
returns
the Jacobian ∂f/∂y or
to the matrix ∂f/∂y if
the Jacobian is constant. If the Jacobian
property
is not set (the default), ∂f/∂y is
approximated by finite differences. Set the Vectorized
property
'on
' if the ODE function is coded so that odefun
(T
,[Y1,Y2
...
]) returns [odefun
(T
,Y1
),odefun
(T
,Y2
),...
].
If ∂f/∂y is a
sparse matrix, set the JPattern
property to the
sparsity pattern of ∂f/∂y,
i.e., a sparse matrix S
with S(i,j) =
1
if the i
th component of f(t,y)
depends on the j
th component of y,
and 0 otherwise.
The solvers of the ODE suite can solve problems of the form M(t,y)y′ = f(t,y),
with time- and state-dependent mass matrix M. (The ode23s
solver
can solve only equations with constant mass matrices.) If a problem
has a mass matrix, create a function M = MASS(t,y)
that
returns the value of the mass matrix, and use odeset
to
set the Mass
property to @MASS
.
If the mass matrix is constant, the matrix should be used as the value
of the Mass
property. Problems with state-dependent
mass matrices are more difficult:
If the mass matrix does not depend on the state variable y and
the function MASS
is to be called with one input
argument, t
, set the MStateDependence
property
to 'none
'.
If the mass matrix depends weakly on y,
set MStateDependence
to 'weak
'
(the default); otherwise, set it to 'strong
'. In
either case, the function MASS
is called with
the two arguments (t
,y
).
If there are many differential equations, it is important to exploit sparsity:
Return a sparse M(t,y).
Supply the sparsity pattern of ∂f/∂y using
the JPattern
property or a sparse ∂f/∂y using
the Jacobian
property.
For strongly state-dependent M(t,y),
set MvPattern
to a sparse matrix S
with S(i,j)
= 1
if for any k
, the (i,k
)
component of M(t,y)
depends on component j
of y,
and 0
otherwise.
If the mass matrix M is singular, then M(t,y)y′ = f(t,y)
is a system of differential algebraic equations. DAEs have solutions
only when y_{0} is consistent,
that is, if there is a vector yp_{0} such
that M(t_{0},y_{0})yp_{0} = f(t_{0},y_{0}).
The ode15s
and ode23t
solvers
can solve DAEs of index 1 provided that y0
is sufficiently
close to being consistent. If there is a mass matrix, you can use odeset
to set the MassSingular
property
to 'yes'
, 'no'
, or 'maybe'
.
The default value of 'maybe'
causes the solver
to test whether the problem is a DAE. You can provide yp0
as
the value of the InitialSlope
property. The default
is the zero vector. If a problem is a DAE, and y0
and yp0
are
not consistent, the solver treats them as guesses, attempts to compute
consistent values that are close to the guesses, and continues to
solve the problem. When solving DAEs, it is very advantageous to formulate
the problem so that M is a diagonal matrix (a semi-explicit
DAE).
Solver | Problem Type | Order of Accuracy | When to Use |
---|---|---|---|
| Nonstiff | Medium | Most of the time. This should be the first solver you try. |
| Nonstiff | Low | For problems with crude error tolerances or for solving moderately stiff problems. |
| Nonstiff | Low to high | For problems with stringent error tolerances or for solving computationally intensive problems. |
| Stiff | Low to medium | If |
| Stiff | Low | If using crude error tolerances to solve stiff systems and the mass matrix is constant. |
| Moderately Stiff | Low | For moderately stiff problems if you need a solution without numerical damping. |
| Stiff | Low | If using crude error tolerances to solve stiff systems. |
The algorithms used in the ODE solvers vary according to order of accuracy [6] and the type of systems (stiff or nonstiff) they are designed to solve. See Algorithms for more details.
Different solvers accept different parameters in the options
list. For more information, see odeset
and Integrator Options in the MATLAB^{®} Mathematics
documentation.
Parameters | ode45 | ode23 | ode113 | ode15s | ode23s | ode23t | ode23tb |
---|---|---|---|---|---|---|---|
| √ | √ | √ | √ | √ | √ | √ |
| √ | √ | √ | √ | √ | √ | √ |
| √ | √ | √ | √ * | — | √ * | √ * |
| √ | √ | √ | √ | √ | √ | √ |
| √ | √ | √ | √ | √ | √ | √ |
| — | — | — | √ | √ | √ | √ |
| √ | √ | √ | √ | √ | √ | √ |
| — | — | — | √ | — | √ | — |
| — | — | — | √ | — | — | — |
Note
You can use the |
An example of a nonstiff system is the system of equations describing the motion of a rigid body without external forces.
$$\begin{array}{cc}\begin{array}{cc}\begin{array}{l}{{y}^{\prime}}_{1}={y}_{2}{y}_{3}\text{}{y}_{1}(0)=0\\ {{y}^{\prime}}_{2}=-{y}_{1}{y}_{3}\text{}{y}_{2}(0)=1\\ \begin{array}{cc}{{y}^{\prime}}_{3}=-0.51{y}_{1}{y}_{2}& {y}_{3}(0)=1\end{array}\end{array}& \end{array}& \end{array}$$
To simulate this system, create a function rigid
containing
the equations
function dy = rigid(t,y) dy = zeros(3,1); % a column vector dy(1) = y(2) * y(3); dy(2) = -y(1) * y(3); dy(3) = -0.51 * y(1) * y(2);
In this example we change the error tolerances using the odeset
command and solve on a time interval [0
12]
with an initial condition vector [0 1 1]
at
time 0
.
options = odeset('RelTol',1e-4,'AbsTol',[1e-4 1e-4 1e-5]); [T,Y] = ode45(@rigid,[0 12],[0 1 1],options);
Plotting the columns of the returned array Y
versus T
shows
the solution
plot(T,Y(:,1),'-',T,Y(:,2),'-.',T,Y(:,3),'.')
An example of a stiff system is provided by the van der Pol equations in relaxation oscillation. The limit cycle has portions where the solution components change slowly and the problem is quite stiff, alternating with regions of very sharp change where it is not stiff.
$$\begin{array}{l}\begin{array}{cc}{{y}^{\prime}}_{1}={y}_{2}& {y}_{1}(0)=2\\ {{y}^{\prime}}_{2}=1000(1-{y}_{1}^{2}){y}_{2}-{y}_{1}& {y}_{2}(0)=0\end{array}\\ \end{array}$$
To simulate this system, create a function vdp1000
containing
the equations
function dy = vdp1000(t,y) dy = zeros(2,1); % a column vector dy(1) = y(2); dy(2) = 1000*(1 - y(1)^2)*y(2) - y(1);
For this problem, we will use the default relative and absolute
tolerances (1e-3
and 1e-6
, respectively)
and solve on a time interval of [0 3000]
with initial
condition vector [2 0]
at time 0
.
[T,Y] = ode15s(@vdp1000,[0 3000],[2 0]);
Plotting the first column of the returned matrix Y
versus T
shows
the solution
plot(T,Y(:,1),'-o')
This example solves an ordinary differential equation with time-dependent terms.
Consider the following ODE, with time-dependent parameters defined only through the set of data points given in two vectors:
y'(t) + f(t)y(t) = g(t)
y(1) = 1
, where the function f(t)
is
defined by the n
-by-1
vector f
evaluated
at times ft
, and the function g(t)
is
defined by the m
-by-1
vector g
evaluated
at times gt
.First, define the time-dependent parameters f(t)
and g(t)
as
the following:
ft = linspace(0,5,25); % Generate t for f f = ft.^2 - ft - 3; % Generate f(t) gt = linspace(1,6,25); % Generate t for g g = 3*sin(gt-0.25); % Generate g(t)
function dydt = myode(t,y,ft,f,gt,g) f = interp1(ft,f,t); % Interpolate the data set (ft,f) at time t g = interp1(gt,g,t); % Interpolate the data set (gt,g) at time t dydt = -f.*y + g; % Evaluate ODE at time t
myode.m
within
the MATLAB ode45
function specifying time
as the first input argument :Tspan = [1 5]; % Solve from t=1 to t=5 IC = 1; % y(t=1) = 1 [T Y] = ode45(@(t,y) myode(t,y,ft,f,gt,g),Tspan,IC); % Solve ODE
y(t)
as a function
of time: plot(T, Y); title('Plot of y as a function of time'); xlabel('Time'); ylabel('Y(t)');
[1] Bank, R. E., W. C. Coughran, Jr., W. Fichtner, E. Grosse, D. Rose, and R. Smith, "Transient Simulation of Silicon Devices and Circuits," IEEE Trans. CAD, 4 (1985), pp. 436–451.
[2] Bogacki, P. and L. F. Shampine, "A 3(2) pair of Runge-Kutta formulas," Appl. Math. Letters, Vol. 2, 1989, pp. 321–325.
[3] Dormand, J. R. and P. J. Prince, "A family of embedded Runge-Kutta formulae," J. Comp. Appl. Math., Vol. 6, 1980, pp. 19–26.
[4] Forsythe, G. , M. Malcolm, and C. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, New Jersey, 1977.
[5] Kahaner, D. , C. Moler, and S. Nash, Numerical Methods and Software, Prentice-Hall, New Jersey, 1989.
[6] Shampine, L. F. , Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.
[7] Shampine, L. F. and M. K. Gordon, Computer Solution of Ordinary Differential Equations: the Initial Value Problem, W. H. Freeman, San Francisco, 1975.
[8] Shampine, L. F. and M. E. Hosea, "Analysis and Implementation of TR-BDF2," Applied Numerical Mathematics 20, 1996.
[9] Shampine, L. F. and M. W. Reichelt, "The MATLAB ODE Suite," SIAM Journal on Scientific Computing, Vol. 18, 1997, pp. 1–22.
[10] Shampine, L. F., M. W. Reichelt, and J.A. Kierzenka, "Solving Index-1 DAEs in MATLAB and Simulink," SIAM Review, Vol. 41, 1999, pp. 538–552.
deval
| function_handle
| ode15i
| odeget
| odeset