# Ode15s

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Scarica una trial. You are now following this question You will see updates in your activity feed. You may receive emails, depending on your notification preferences. Vote 0. Accepted Answer: Andrew Newell. Solving a partial differential equation with ode15s,we know that this ode solver integrates the ode over time direvatives. I need to find the values of these time direvatives,do you have any special command? Accepted Answer.Create or alter options structure for input to ordinary differential equation ODE solvers.

The ODE solvers can integrate systems of differential equations of one of these forms. Any unspecified properties have default values. It is sufficient to type only the leading characters that uniquely identify a property name.

Case is ignored for property names. Any new options not equal to the empty matrix overwrite corresponding options in oldopts. There are several categories of properties:. The Version 5 properties are supported only for backward compatibility. If scalar, the tolerance applies to all components of the solution vector. Otherwise the tolerances apply to corresponding components.

By default the solvers use a more stringent component-wise error control. OutputSel Vector of integers Output selection indices. Specifies the components of the solution vector that the solver passes to the output function.

OutputSel defaults to all components. Refine Positive integer Produces smoother output, increasing the number of output points by the specified factor. The default value is 1 in all solvers except ode45where it is 4. Set this property to FJac if a function FJac t,y returns or to the constant value of. Set this property to a sparse matrix with if component of depends on component ofand 0 otherwise.

That is, your ODE function can pass to the solver a whole array of column vectors at once. A stiff function calls your ODE function in a vectorized manner only if it is generating Jacobians numerically the default behavior and you have used odeset to set Vectorized to on. Set this property to Eventswhere Events is the name of the events function.Arguments odefun A function that evaluates the right-hand side of the differential equations. All solvers solve systems of equations in the form or problems that involve a mass matrix.

The ode23s solver can solve only equations with constant mass matrices. See odeset for details. Optional parameters that the solver passes to odefun and all the functions specified in options. Each row in the solution array Y corresponds to a time returned in column vector T. Commonly used properties include a scalar relative error tolerance RelTol 1e-3 by default and a vector of absolute error tolerances AbsTol all componen t s are 1e-6 by default.

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 'Events' property to a function, e. For the i th event function in events :. Corresponding entries in TEYEand 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. You must pass odefun as a function handle. The structure sol always includes these fields:. If you specify the Events option and events are detected, sol also includes these fields:. The values indicate which event the solver detected.

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: odeplotodephas2odephas3odeprint. When you call the solver with no output arguments, it calls the default odeplot to plot the solution as it is computed. 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 ode15sode23sode23tand ode23tbthe Jacobian matrix is critical to reliability and efficiency.

If the Jacobian property is not set the defaultis approximated by finite differences. If is a sparse matrix, set the JPattern property to the sparsity pattern ofi. The solvers of the ODE suite can solve problems of the formwith time- and state-dependent mass matrix.

Stiff Systems & Solution using ode15s

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 there are many differential equations, it is important to exploit sparsity:. If the mass matrix is singular, then is a differential algebraic equation.

DAEs have solutions only when is consistent, that is, if there is a vector such that. The ode15s and ode23t solvers can solve DAEs of index 1 provided that y0 is sufficiently close to being consistent.Create or alter options structure for input to ordinary differential equation ODE solvers. In the resulting structure, optionsthe named properties have the specified values.

For example, 'name1' has the value value1. Any unspecified properties have default values. It is sufficient to type only the leading characters that uniquely identify a property name. Case is ignored for property names. The modified structure is returned as an output argument. Any new options not equal to the empty matrix overwrite corresponding options in oldopts.

ODE Properties The following sections describe the properties that you can set using odeset. The available properties depend on the ODE solver you are using.

There are several categories of properties:. The Version 5 properties are supported only for backward compatibility. Error Control Properties At each step, the solver estimates the local error e in the i th component of the solution. This error must be less than or equal to the acceptable error, which is a function of the specified relative tolerance, RelToland the specified absolute tolerance, AbsTol. For routine problems, the ODE solvers deliver accuracy roughly equivalent to the accuracy you request.

They deliver less accuracy for problems integrated over "long" intervals and problems that are moderately unstable. Difficult problems may require tighter tolerances than the default values. For relative accuracy, adjust RelTol. For the absolute error tolerance, the scaling of the solution components is important: if y is somewhat smaller than AbsTolthe solver is not constrained to obtain any correct digits in y.

You might have to solve a problem more than once to discover the scale of solution components. Roughly speaking, this means that you want RelTol correct digits in all solution components except those smaller than thresholds AbsTol i. Even if you are not interested in a component y i when it is small, you may have to specify AbsTol i small enough to get some correct digits in y i so that you can accurately compute more interesting components.

The following table describes the error control properties. Solver Output Properties The following table lists the solver output properties, which control the output that the solvers generate. To specify an output function, set 'OutputFcn' to a function handle. The solver calls the specified output function with the following flags.

Note that the syntax of the call differs with the flag. The function must respond appropriately: init -- The solver calls myfun tspan,y0,'init' before beginning the integration to allow the output function to initialize. If t is a vector, the i th column of y corresponds to the i th element of t.A problem is said to be stiff if the solution being sought varies slowly, but there are nearby solutions that vary rapidly, so the numerical method must take small steps to obtain satisfactory results.

The flame model demonstrates stiffness. These methods require more work per step, but take many fewer steps. The differential equation is y prime is y squared minus y cubed, and I'm going to choose a fairly-- an extremely small initial condition, 10 to the minus sixth. The final value of t is 2 over y naught, and I'm going to impose a modest accuracy requirement, 10 to the minus fifth. Now let's run ode45 with its default output. Now, see it's taking-- it's moving very slowly here.

It's taking lots of steps. So I'm take- pressing the stop button here. It's working very hard.

### Solving ODEs in MATLAB, 7: Stiffness, ODE23s, ODE15s

Let's zoom in and see why it's taking so many steps, very densely packed steps here. This is stiffness. It's satisfying the accuracy requirements we imposed. All these steps are within 10 to the minus sixth of one, but it's taken very small steps to do it. These steps are so small that the graphics can't even discern the step size. It's an efficiency issue.

It's doing what we asked for. It's meeting the accuracy requirements, but it's having to take very small steps to do it. Let's try another ODE solver-- ode Just change this to 23 and see what it does.

It's also taking very small steps for the same reason. If we zoom in on here, we'll see the same kind of behavior. But it's taking very small steps in order to achieve the desired accuracy.

Now let me introduce a new solver, ode23s. The s for stiffness.

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This was designed to solve stiff problems.Documentation Help Center. Each row in the solution array y corresponds to a value returned in column vector t. The solvers all use similar syntaxes. The ode23s solver only can solve problems with a mass matrix if the mass matrix is constant. Specify the mass matrix using the Mass option of odeset. For example, use the AbsTol and RelTol options to specify absolute and relative error tolerances, or the Mass option to provide a mass matrix.

In the output, te is the time of the event, ye is the solution at the time of the event, and ie is the index of the triggered event. For each event function, specify whether the integration is to terminate at a zero and whether the direction of the zero crossing matters. You can use any of the input argument combinations in previous syntaxes.

Simple ODEs that have a single solution component can be specified as an anonymous function in the call to the solver. The anonymous function must accept two inputs t,y even if one of the inputs is not used. An example of a stiff system of equations is the van der Pol equations in relaxation oscillation. The limit cycle has regions where the solution components change slowly and the problem is quite stiff, alternating with regions of very sharp change where it is not stiff.

The initial conditions are and. Solving this system using ode45 with the default relative and absolute error tolerances 1e-3 and 1e-6respectively is extremely slow, requiring several minutes to solve and plot the solution.

This is a plot of the solution obtained by ode45which takes a long time to compute. Notice the enormous number of time steps required to pass through areas of stiffness. Solve the stiff system using the ode15s solver, and then plot the first column of the solution y against the time points t. The ode15s solver passes through stiff areas with far fewer steps than ode However, you can pass in extra parameters by defining them outside the function and passing them in when you specify the function handle.

Solve the ODE using ode15s. Specify the function handle such that it passes in the predefined values for A and B to odefcn. The ode15s solver is a good first choice for most stiff problems.

However, the other stiff solvers might be more efficient for certain types of problems. This example solves a stiff test equation using all four stiff ODE solvers.

With these values, the problem is stiff enough that ode45 and ode23 struggle to integrate the equation. Solve the equation with ode15sode23sode23tand ode23tb. Make subplots for comparison. The stiff solvers all perform well, but ode23s completes the integration with the fewest steps and runs the fastest for this particular problem.

Since the constant Jacobian is specified, none of the solvers need to calculate partial derivatives to compute the solution. Specifying the Jacobian benefits ode23s the most since it normally evaluates the Jacobian in every step. For general stiff problems, the performance of the stiff solvers varies depending on the format of the problem and specified options. Providing the Jacobian matrix or sparsity pattern always improves solver efficiency for stiff problems.

But since the stiff solvers use the Jacobian differently, the improvement can vary significantly. Practically speaking, if a system of equations is very large or needs to be solved many times, then it is worthwhile to investigate the performance of the different solvers to minimize execution time.Solve differential equations Syntax. Go to function:. All solvers can solve systems of equations in the form.

Of these four solvers all but ode23s can solve equations in the form. For information about ODE file syntax, see the odefile reference page. A vector specifying the interval of integration [t0 tfinal].

Optional integration argument created using the odeset function. See odeset for details. Solution matrix Ywhere each row corresponds to a time returned in column vector T. If using stringent error tolerances or solving a computationally intensive ODE file.

If ode45 is slow stiff systems or there is a mass matrix. If using crude error tolerances to solve stiff systems or there is a constant mass matrix. If the problem is only moderately stiff and you need a solution without numerical damping. If using crude error tolerances to solve stiff systems or there is a mass matrix.

The algorithms used in the ODE solvers vary according to order of accuracy  and the type of systems stiff or nonstiff they are designed to solve. See Algorithms for more details. It is possible to specify tspany0 and options in the ODE file see odefile. Empty arguments at the end of the call list may be omitted. If an option is specified in both places, the command line specification takes precedence. For information about constructing an ODE file, see the odefile reference page.

Options Different solvers accept different parameters in the options list.