For example, if the equations have stiffness, implicit methods will be used as needed, or if the equations make a DAE, a special DAE method will be used. With the default setting Method-> Automatic, NDSolve will choose a method which should be appropriate for the differential equations. NDSolve has several different methods built in for computing solutions as well as a mechanism for adding additional methods. With the setting MaxSteps-> Infinity there is no upper limit on the number of steps used. When solutions have a complicated structure, however, you may sometimes have to choose larger settings for MaxSteps. The default setting MaxSteps->10000 should be sufficient for most equations with smooth solutions. The norm is scaled in terms of the tolerances, given so that NDSolve tries to take steps such that When working with systems of equations, it uses the setting of the option NormFunction-> f to combine errors in different components. NDSolve uses error estimates for determining whether it is meeting the specified tolerances. With the default setting of Automatic, both AccuracyGoal and PrecisionGoal are equal to half of the setting for WorkingPrecision. If you specify large values for AccuracyGoal or PrecisionGoal, then you typically need to give a somewhat larger value for WorkingPrecision. NDSolve uses the setting you give for WorkingPrecision to determine the precision to use in its internal computations. For some differential equations, this error can accumulate, so it is possible that the precision or accuracy of the result at the end of the time interval may be much less than what you might expect from the settings of AccuracyGoal and PrecisionGoal. Generally, AccuracyGoal and PrecisionGoal are used to control the error local to a particular time step. By setting AccuracyGoal-> Infinity, you tell NDSolve to use PrecisionGoal only. If you need to track a solution whose value comes close to zero, then you will typically need to increase the setting for AccuracyGoal. The setting for AccuracyGoal effectively determines the absolute error to allow in the solution, while the setting for PrecisionGoal determines the relative error. In general, NDSolve makes the steps it takes smaller and smaller until the solution reached satisfies either the AccuracyGoal or the PrecisionGoal you give. NDSolve allows you to specify the precision or accuracy of result you want. In general, if the solution appears to be varying rapidly in a particular region, then NDSolve will reduce the step size to be able to better track the solution. NDSolve uses an adaptive procedure to determine the size of these steps. are solvable for the highest derivative order), but only linear boundary value problems.Īs mentioned earlier, NDSolve works by taking a sequence of steps in the independent variable t. NDSolve can solve nearly all initial value problems that can symbolically be put in normal form (i.e. A boundary value occurs when there are multiple points t. When there is only one t at which conditions are given, the equations and initial conditions are collectively referred to as an initial value problem. These conditions specify values for u i, and perhaps derivatives u i', at particular points t. In order to get started, NDSolve has to be given appropriate initial or boundary conditions for the u i and their derivatives. It starts at a particular value of t, then takes a sequence of steps, trying eventually to cover the whole range t min to t max. In general, NDSolve finds solutions iteratively. The InterpolatingFunction objects provide approximations to the u i over the range of values t min to t max for the independent variable t. NDSolve represents solutions for the functions u i as InterpolatingFunction objects. NDSolve įind numerical solutions for several functions u iįinding numerical solutions to ordinary differential equations.
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