How to change the numerical integration method

It may be necessary to change the solver depending on the problem. Some of the numerical integration solvers available as MATLAB standard functions can be selected in BunKi.

1. Select Setting -> Solver.

2. Start up Solver option window.

3. Select the appropriate numerical integration solver from the pull down menu in the Solver field.

• See the following for details of each solver.
 ode45 This is based on an explicit Runge-Kutta formula, the Dormand-Prince pair. It is a one-step solver; that is, in computing y(tn), it needs only the solution at the immediately preceding time point, y(tn-1). In general, ode45 is the best solver to apply as a "first try" for most problems. ode23 This is also based on an explicit Runge-Kutta proposed by Bogacki and Shampine. It may be more efficient than ode45 at crude tolerances and in the presence of mild stiffness. ode23 is a one-step solver. ode113 This is a variable order Adams-Bashforth-Moulton PECE solver. It may be more efficient than ode45 at stringent tolerances. ode113 is a multi-step solver; that is, it normally needs the solutions at several preceding time points to compute the current solution. ode23s}; This is based on a modified Rosenbrock formula of order 2. Because it is a one-step solver, it may be more efficient than ode15s at crude tolerances. It can solve some kinds of stiff problems for which ode15s is not effective. ode15s This is a variable order solver based on the numerical differentiation formulas (NDFs). These are related to but are more efficient than the backward differentiation formulas, BDFs (also known as Gear's method). Like ode113, ode15s is a multi-step method solver. If you suspect that a problem is stiff or if ode45 failed or was very inefficient, try ode15s. myself This is a standard fixed step 4th order Runge Kutta method．It is not recommended unless under particular circumstances.

How to improve the precision of numerical integration

When the precision of the numerical integration solver is bad, it may have harmful effects on the numerical calculation. In this case, the precision of the solver needs to be changed. This can be accomplished by changing the RelTol and AbsTol, measures of tolerance of the error allowed from the real solution.

1. Select Setting -> Solver.

2. Change the Relative Tolerance and Absolute Tolerance parameters.

• The meaning of these parameters are as follows:
• Relative toleranceRelative tolerated error applicable to all vector elements．
• Absolute toleranceAbsolute tolerated error. For scalars, applicable to all solution vector elements. For vectors, applicable to corresponding elements.

How to numerically integrate with a fixed step size

The variable step size algorithm is implemented in the numerical integration solver provided by MALAB that changes the step size according to the state of the orbit of a solution to increase calculation efficiency*1. However, for some problems, a fixed step size may work better. Fixed step size can be used in a numerical integration solver by following the following steps.

1. Select Setting -> Solver.

2. Start up Solver option window.

3. Select Invariable as Step type.

4. Configure Initial step. The initial step is originally solver parameter that the solver uses at the first step (for details, see step size.). Input an appropriate value. This value is the fixed step size.

The response of PP as a whole is bad

The numerical integration solver calculates over a specified time frame and outputs the results. In a real time simulation, calculations preferably should run without specifying an end time. Therefore, PP tool repeatedly runs the processes such that the solver integrates numerically until an internal end-time, displays the result, executes key events, and then resumes calculating. In general, the end time is set the period of limit cycles and periodic solutions. However, for limit cycles observed in an autonomous systems, in some problems, limit cycles with very long periods may have to be displayed. For those cases, if the end time is set the period of the limit cycle, the response of PP tool may become inaccurate, causing some problems.

Therefore, the Solver frequency is set instead of the period of the limit cycle. This indicates that after the solver calculates the amount of configured value at Solver frequency, dispay and key events happen. (If the Poincare section is crossed, the display and key events happen at this point).

Therefore, the Solver frequency value affects the key response, click response, events and the amount to be drawn at one time. There is not much difference if this frequency is increased (value is decreased) in systems with a slow response. For systems with a fast response, if events are to be inserted during the attractor, this value has to be decreased.

*1 In the numerical integration solver, the step size is adjested automatically such that the small step size is selected when the orbit of a solution is abruptly changed while the large step size is selected if the orbit is gradually changed.

Last-modified: 2009-07-23 (Thu) 20:13:30 (3767d)