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#Ndsolve mathematica code#

The construction of compound integration methods is particularly useful in geometric numerical integration. NDSolve ⟶ "Extrapolation" ⟶ "ExplicitMidpoint" This was later extended to allow one method to call another in a sequential fashion, with an arbitrary number of levels of nesting. NDSolve ⟶ "ImplicitRungeKutta" First Revision The original method framework design allowed a number of methods to be invoked in the solver. This is a generalization of the ideas used in the Generic ODE Solving System, Godess (see, , and the references therein), which is implemented in C++. Methods are reentrant and hierarchical, meaning that one method can call another. This is a generalization of the ideas used in codes like LSODA (, ). This includes, but is not limited to, coefficients, workspaces, step-size control parameters, step-size acceptance/rejection information, and Jacobian matrices.

Rounding error feedback (compensated summation) is particularly advantageous for high-order methods or methods that conserve specific quantities during the numerical integrationĮach method has its own data object that contains information that is needed for the invocation of the method.

Unrecoverable integration errors (e.g.

Divergence of iterations in implicit methods (e.g.

Specified (or detected) singularities are handled by restarting the integration.

Step sizes are not decreased drastically toward the end of an integration.

Step sizes are not increased after a step rejection.Step sizes do not change sign unexpectedly, which may be a consequence of user programming error.Step sizes in a numerical integration do not become too small in value, which may happen in solving stiff systems.The routine handles a number of different criteria including: Plug-in capabilities that allow user extensibility and prototypingĪ common time-stepping mechanism is used for all one-step methods.Type and precision dynamic for all methods.Tensor framework that allows families of methods to share one implementation.Vectorized framework based on a generalization of the BLAS model using optimized in-place arithmetic.Uniform treatment of rounding errors (see, , and the references therein).Hierarchical and reentrant numerical methods.Separation of method initialization phase and run-time computation.Objection orientation (method property specification and communication).The principal features of the NDSolve framework are:

#Ndsolve mathematica code#

This approach also allows code optimization to be carried out in just a few central routines.

In order to cut down on maintenance and duplication of code, common components are shared between methods. Supporting a large number of numerical integration methods for differential equations is a lot of work. Automatic Selection and User Controllability