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You can control the accuracy and precision to which FindRoot tries to get the root of the event function using method options for the Brent event location method. The default is to find the root to the same accuracy and precision as NDSolve is using for local error control. For methods that support continuous or dense output, the argument for the event function can be found quite efficiently simply by using the continuous output formula.
However, for methods that do not support continuous output, the solution needs to be computed by taking a step of the underlying method, which can be relatively expensive. An alternate way of getting a solu- tion approximation that is not accurate to the method order, but is consistent with using FindRoot on the InterpolatingFunction object returned from NDSolve is to use cubic Her- mite interpolation, obtaining approximate solution values in the middle of the step by interpola- tion based on the solution values and solution derivative values at the step ends.
This example integrates the pendulum equation for a number of different event location meth- ods and compares the time when the event is found. This defines the event location methods to use. This integrates the system and prints out the method used and the value of the independent variable when the integration is terminated. This system models a body falling under the force of gravity encountering air resistance see [M04]. The event action stores the time when the falling body hits the ground and stops the integration.
This plots the solution and highlights the initial and final points green and red by encircling them. The event locator method can call any method for actually doing the integration of the ODE system. The default method, Automatic, automatically switches to a method appropriate for stiff systems when necessary, so that stiffness does not present a problem.
By selecting the endpoint of the NDSolve solution, it is possible to write a function that returns the period as a function of m.
Using Poincar sections is a useful technique for visualizing the solutions of high-dimensional differential systems.
This means that NDSolve will produce no InterpolatingFunction as output, avoiding storing a lot of unnecessary data. NDSolve does give a message NDSolve::noout warning there will be no output functions, but it can safely be turned off in this case since the data of interest is collected from the event actions.
The linear interpolation event location method is used because the purpose of the computation here is to view the results in a graph with relatively low resolution. If you were doing an exam- ple where you needed to zoom in on the graph to great detail or to find a feature, such as a fixed point of the Poincar map, it would be more appropriate to use the default location method.
This turns off the message warning about no output. This integrates the Hnon Heiles system using a fourth-order explicit Runge Kutta method with fixed step size of 0. The event action is to use Sow on the values of Y2 and Y4.
This plots the Poincar section. The collected data is found in the last part of the result of Reap and the list of points is the first part of that. Since the Hnon Heiles system is Hamiltonian, a symplectic method gives much better qualita- tive results for this example. This integrates the Hnon Heiles system using a fourth-order symplectic partitioned Runge Kutta method with fixed step size of 0.
This loads an example problem of the Arnold Beltrami Childress ABC flow that is used to model chaos in laminar flows of the three-dimensional Euler equations. This defines a splitting Y1, Y2 of the system by setting some of the right-hand side components to zero. This defines the implicit midpoint method. This constructs a second-order splitting method that retains volume and reversing symmetries. This defines a function that gives the Poincar section for a particular initial condition.
This finds the Poincar sections for several different initial conditions and flattens them together into a single list of points. This example is a generalization of an example in [SGT03]. It models a ball bouncing down a ramp with a given profile. The example is good for demonstrating how you can use multiple invocations of NDSolve with event location to model some behavior. This defines a function that computes the solution from one bounce to the next.
The solution is computed until the next time the path intersects the ramp. This defines the function for the bounce when the ball hits the ramp.
The formula is based on reflection about the normal to the ramp assuming only the fraction k of energy is left after a bounce. This defines the function that runs the bouncing ball simulation for a given reflection ratio, ramp, and starting position. This is the example that is done in [SGT03]. The ramp is now defined to be a quarter circle. This adds a slight waviness to the ramp. This example illustrates the solution of the restricted three-body problem, a standard nonstiff test system of four equations.
The example traces the path of a spaceship traveling around the moon and returning to the earth see p. The ability to specify multiple events and the direction of the zero crossing is important. The initial conditions have been chosen to make the orbit periodic. The value of m corresponds to a spaceship traveling around the moon and the earth. The event function is the derivative of the distance from the initial conditions.
A local maximum or minimum occurs when the value crosses zero. There are two events, which for this example are the same. The first event with Direction 1 corresponds to the point where the distance from the initial point is a local minimum, so that the spaceship returns to its original position. The event action is to store the time of the event in the variable tfinal and to stop the integration. The second event corresponds to a local maximum.
The event action is to store the time that the spaceship is farthest from the starting position in the variable tfar. The first two solution components are coordinates of the body of infinitesimal mass, so plotting one against the other gives the orbit of the body.
This displays one half-orbit when the spaceship is at the furthest point from the initial position. This displays one complete orbit when the spaceship returns to the initial position. Collect the data for a local maximum of each component as the integration proceeds. A sepa- rate tag for Sow and Reap is used to distinguish the components.
Display the local maxima together with the solution components. In many applications the function in a differential system may not be analytic or continuous everywhere. In order to illustrate the difficulty in crossing a discontinuity, consider the following example [G84] see also [HNW93] :.
Here is the input for the entire system. The switching function is assigned to the symbol event, and the function defining the system depends on the sign of the switching function. The symbol odemethod is used to indicate the numerical method that should be used for the integration. For comparison, you might want to define a different method, such as ExplicitRungeKutta, and rerun the computations in this section to see how other meth- ods behave. This solves the system on the interval [0, 1] and collects data for the mesh points of the integra- tion using Reap and Sow.
Despite the fact that a solution has been obtained, it is not clear whether it has been obtained efficiently. The following example shows that the crossing of the discontinuity presents difficulties for the numerical solver.
This defines a function that displays the mesh points of the integration together with the num- ber of integration steps that are taken. As the integration passes the discontinuity near 0. One of the most efficient methods of crossing a discontinuity is to break the integration by restarting at the point of discontinuity. The following example shows how to use the EventLocator method to accomplish this. This numerically integrates the first part of the system up to the point of discontinuity.
The switching function is given as the event. The direction of the event is restricted to a change from negative to positive. When the event is found, the solution and the time of the event are stored by the event action. Using the discontinuity found by the EventLocator method as a new initial condition, the integration can now be continued.
This defines a system and initial condition, solves the system numerically, and collects the data used for the mesh points. A plot of the two solutions is very similar to that obtained by solving the entire system at once. Examining the mesh points, it is clear that far fewer steps were taken by the method and that the problematic behavior encountered near the discontinuity has been eliminated.
The value of the discontinuity is given as 0. In this example it is possible to analytically solve the system and use a numerical method to check the value. The solution of the system up to the discontinuity can be represented in terms of Bessel and gamma functions. Substituting in the solution into the switching function, a local minimization confirms the value of the discontinuity.
Many evolution equations model behavior on a spatial domain that is infinite or sufficiently large to make it impractical to discretize the entire domain without using specialized discretization methods.
In practice, it is often the case that it is possible to use a smaller computational domain for as long as the solution of interest remains localized. In situations where the boundaries of the computational domain are imposed by practical consid- erations rather than the actual model being studied, it is possible to pick boundary conditions appropriately.
Using a pseudospectral method with periodic boundary conditions can make it possible to increase the extent of the computational domain because of the superb resolution of the periodic pseudospectral approximation. The drawback of periodic boundary conditions is that signals that propagate past the boundary persist on the other side of the domain, affecting the solution through wraparound.
It is possible to use an absorbing layer near the boundary to minimize these effects, but it is not always possible to completely eliminate them. The sine-Gordon equation turns up in differential geometry and relativistic field theory.
This example integrates the equation, starting with a localized initial condition that spreads out. The periodic pseudospectral method is used for the integration. Since no absorbing layer has been instituted near the boundaries, it is most appropriate to stop the integration once wraparound becomes significant.
This condition is easily detected with event location using the EventLocator method. The integration is stopped when the size of the solution at the periodic wraparound point crosses a threshold of 0.
This extracts the ending time from the InterpolatingFunction object and makes a plot of the computed solution. You can see that the integration has been stopped just as the first waves begin to reach the boundary. The DiscretizedMonitorVariables option affects the way the event is interpreted for PDEs; with the setting True , u t, xD is replaced by a vector of discretized values.
This is much more efficient because it avoids explicitly constructing the InterpolatingFunction to evaluate the event. The following example constructs a table making a comparison for two different integration methods. This defines a function that returns the time it takes to compute a solution of a mildly damped pendulum equation up to the point at which the bob has momentarily been at rest times. This uses the function to make a table comparing the different location methods for two differ- ent ODE integration methods.
It is worth noting that, often, a significant part of the extra time for computing events arises from the need to evaluate the event functions at each time step to check for the possibility of a sign change.
An optimization is performed for event functions involving only the independent variable. Such events are detected automatically at initialization time. Limitations One limitation of the event locator method is that since the event function is only checked for sign changes over a step interval, if the event function has multiple roots in a step interval, all or some of the events may be missed.
This typically only happens when the solution to the ODE varies much more slowly than the event function. When you suspect that this may have occurred, the simplest solution is to decrease the maximum step size the method can take by using the MaxStepSize option to NDSolve. More sophisticated approaches can be taken, but the best approach depends on what is being computed. An example follows that demonstrates the problem and shows two approaches for fixing it. This should compute the number of positive integers less than 5 there are However, most are missed because the method is taking large time steps because the solution x tD is so simple.
It is quite apparent from the nature of the example problem that if the endpoint is increased, it is likely that a smaller maximum step size may be required. Taking very small steps every- where is quite inefficient.
It is possible to introduce an adaptive time step restriction by setting up a variable that varies on the same time scale as the event function. This introduces an additional function to integrate that is the event function.
Settings for the EventLocationMethod option. Introduction Extrapolation methods are a class of arbitrary-order methods with automatic order and step- size control. The error estimate comes from computing a solution over an interval using the same method with a varying number of steps and using extrapolation on the polynomial that fits through the computed solutions, giving a composite higher-order method [BS64].
At the same time, the polynomials give a means of error estimation. Typically, for low precision, the extrapolation methods have not been competitive with Runge Kutta-type methods. For high precision, however, the arbitrary order means that they can be arbitrarily faster than fixed-order methods for very precise tolerances. The order and step-size control are based on the codes odex. This loads packages that contain some utility functions for plotting step sequences and some predefined problems.
Extrapolation generalizes the idea of Richardson's extrapolation to a sequence of refine- ments. Choose a numerical method of order p and compute the solution of the initial value problem by carrying out ni steps with step size hi to obtain:. Extrapolation is performed using the Aitken Neville algorithm by building up a table of values:. For stiff problems the analysis is. Any extrapolation sequence can be specified in the implementation.
Some common choices are as follows. This is the Romberg sequence. This is the Bulirsch sequence. This is the harmonic sequence. For a base method of order two, the first entries in the sequence are given by the following. Here is an example of adding a function to define the harmonic sequence where the method order is an optional pattern.
The sequence with lowest cost is the Harmonic sequence, but this is not without problems since rounding errors are not damped. For high-order extrapolation an important consideration is the accumulation of rounding errors. Suppose that the entries T11 , T21 , T31 , are disturbed with rounding errors e, -e, e, and com- pute the propagation of these errors into the extrapolation table. Due to the linearity of the extrapolation process 2 , suppose that the Ti, j are equal to zero and.
Hence, for an order-sixteen method approximately two decimal digits are lost due to rounding error accumulation. It seems worthwhile to look for approaches that can reduce the effect of rounding errors in high- order extrapolation. Selecting a different step sequence to diminish rounding errors is one approach, although the drawback is that the number of integration steps needed to form the Ti,1 in the first column of the extrapolation table requires more work.
Some codes, such as STEP, take active measures to reduce the effect of rounding errors for stringent tolerances [SG75]. An alternative strategy, which does not appear to have received a great deal of attention in the context of extrapolation, is to modify the base-integration method in order to reduce the magni- tude of the rounding errors in floating-point operations.
This approach, based on ideas that dated back to [G51], and used to good effect for the two-body problem in [F96b] for back- ground see also [K65], [M65a], [M65b], [V79] , is explained next. Base Methods The following methods are the most common choices for base integrators in extrapolation. For efficiency, these have been built into NDSolve and can be called via the Method option as individual methods. The implementation of these methods has a special interpretation for multiple substeps within DoubleStep and Extrapolation.
The NDSolve. This is advantageous for geometric numerical integration where numeri- cal errors are not damped over long time integrations. It also allows the application of efficient correction strategies such as compensated summation. This formulation is also useful in the context of extrapolation. The methods are now described together with the increment reformulation that is used to reduce rounding error accumulation.
It is well-known that, for certain base integration schemes, the entries Ti, j in the extrapolation. Then the integration 1 can be rewritten to reflect the correspondence with.
Mathematically the formulations 1 and 1, 2 are equivalent. Expansions in even powers of h are extremely important for an efficient implementation of Richardson's extrapolation and an elegant proof is given in [S70]. The smoothing step of Gragg has its historical origins in the weak stability of the explicit mid- point rule:.
In order to make use of 1 , the formulation 1 is computed with 2 nk steps. This has the advan-. Notice that because of the construction, a sum of increments is available at the end of the algorithm together with two consecutive increments.
This leads to the following formulation:. Gragg's smoothing step is not of great importance if the method is followed by extrapolation, and Shampine proposes an alternative smoothing procedure that is slightly more efficient [SB83].
The following figures illustrate the effect of the smoothing step on the linear stability domain carried out using the package FunctionApproximations. Since the precise stability boundary can be complicated to compute for an arbitrary base method, a simpler approximation is used. For an extrapolation method of order p, the intersec- tion with the negative real axis is considered to be the point at which:.
The stabillity region is approximated as a disk with this radius and origin 0,0 for the negative half-plane. Increments arise naturally in the description of many semi-implicit and implicit methods. Con- sider a succession of integration steps carried out using the linearly implicit Euler method for. Reformulation in terms of increments as departures from y0 can be accomplished as follows:.
If 1 is computed for 2 nk - 1 linearly implicit midpoint steps, then the method has a symmetric. The smoothing step for the linearly implicit midpoint rule has a different role from Gragg's smoothing for the explicit midpoint rule see [BD83] and [SB83]. Since there is no weakly stable term to eliminate, the aim is to improve the asymptotic stability. You have seen how to modify Ti,1 , the entries in the first column of the extrapolation table, in terms of increments.
However, for certain base integration methods, each of the Ti, j corresponds to an explicit Runge. Therefore, it appears that the correspondence has not yet been fully exploited and further refinement is possible. Since the Aitken Neville algorithm 2 involves linear differences, the entire extrapolation pro-. The advantage is that the extrapolation table is built up using smaller quantities, and so the effect of rounding errors from subtractive cancellation is reduced. Implementation Issues There are a number of important implementation issues that should be considered, some of which are mentioned here.
The Jacobian is evaluated only once for all entries Ti,1 at each time step by storing it together with the associated time that it is evaluated. This also has the advantage that the Jacobian does not need to be recomputed for rejected steps. In order to adaptively change the order of the extrapolation throughout the integration, it is important to have a measure of the amount of work required by the base scheme and extrapola- tion sequence. The dimension of the system, preferably with a weighting according to structure, needs to be incorporated for linearly implicit schemes in order to take account of the expense of solving each linear system.
Two forms of stability check are used for the linearly implicit base schemes for further discus- sion, see [HW96]. In order to interrupt computations in the computation of T1,1 , Deuflhard suggests checking if the Newton iteration applied to a fully implicit scheme would converge. Notice that 1 differs from 1 only in the second equation. It requires finding the solution for a. Increments are a more accurate formulation for the implementation of both forms of stability check.
For comparing different extrapolation schemes, consider an example from [HW96]. This example involves an eighth-order extrapolation of ExplicitEuler with the harmonic sequence. Approximately two digits of accuracy are gained by using the increment-based formu- lation throughout the extrapolation process. The results for the increment formulation 1 followed by standard extrapolation 2 are depicted in blue.
The results for the increment formulation 1 with extrapolation carried out on the incre- ments using 1 are depicted in red.
Plot of work vs error on a log-log scale -8 1. Approximately two decimal digits of accuracy are gained by using the increment-based formula- tion throughout the extrapolation process.
This compares the relative error in the integration data that forms the initial column of the extrapolation table for the previous example. Notice that the rounding-error model that was used to motivate the study of rounding-error growth is limited because in practice, errors in Ti,1 can exceed 1 ULP. The increment formulation used throughout the extrapolation process produces rounding errors in Ti,1 that are smaller than 1 ULP.
This compares the work required for extrapolation based on ExplicitEuler red , the ExplicitMidpoint blue , and ExplicitModifiedMidpoint green. Plot of work vs error on a log-log scale 23 19 15 11 7 1.
Select a problem to solve. Define a monitor function to store the order and the time of evaluation. Use the monitor function to collect data as the integration proceeds. Display how the order varies during the integration. Select the problem to solve. A reference solution is computed with a method that switches between a pair of Extrapolation methods, depending on whether the problem appears to be stiff. Define a list of methods to compare. The work-error comparison data for the methods is displayed in the following logarithmic plot, where the global error is displayed on the vertical axis and the number of function evaluations on the horizontal axis.
Eventually the higher order of the extrapolation methods means that they are more efficient. Note also that the increment formulation continues to give good results even at very stringent tolerances. One of the simplest nonlinear equations describing a circuit is van der Pol's equation. This solves the equations using Extrapolation with the ExplicitModifiedMidpoint base method with the default double-harmonic sequence 2, 4, 6,. The stiffness detection device terminates the integration and an alternative method is suggested.
This solves the equations using Extrapolation with the LinearlyImplicitEuler base method with the default sub-harmonic sequence 2, 3, 4,. Notice that the Jacobian matrix is computed automatically user-specifiable by using either numerical differences or symbolic derivatives and appropriate linear algebra routines are selected and invoked at run time.
This plots the first solution component over time. This plots the step sizes taken in computing the solution. Select the Lorenz equations. This invokes a bigfloat, or software floating-point number, embedded explicit Runge Kutta method of order 9 8 [V78].
The maximum order of these methods is twelve. This invokes the Extrapolation method with ExplicitModifiedMidpoint as the base integration scheme.
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