By Jenna Brandenburg, Lashaun Clemmons
This publication presents a normal method of research of Numerical Differential Equations and Finite aspect procedure
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Extra info for Analysis of numerical differential equations and finite element method
Using the equation given, we obtain the following: which means that h must be smaller than the above to get the desired error or less, and 4 / h, or about 1072 iterations will need to be completed to do so. The large number of steps, and thus high computation cost, supports the use of alternative, higher-order methods such as Runge-Kutta methods or linear multistep methods Numerical stability The Euler method can also be numerically unstable, especially for stiff equations. This limitation—along with its slow convergence of error with h—means that the Euler method is not often used, except as a simple example of numerical integration.
Compact stencils may be compared to non-compact stencils. Compact stencils are currently implemented in many partial differential equation solvers, including several in the topics of CFD, FEA, and other mathematical solvers relating to PDE's. Two Point Stencil Example The two point stencil for the first derivative of a function is given by: . This is obtained from the Taylor series expansion of the first derivative of the function given by: . Replacing h with − h, we have: . Addition of the above two equations together results in the cancellation of the terms in odd powers of h: .
The error bound on the global error is given by: where h is the step size, M is the upper bound on the second derivative of y on the given interval (which must be estimated), and L is the Lipschitz constant. If the error bound is computed, it can be seen, once again, that if small error is desired, the step size h must be very small. 01, assuming a maximum value for the second derivative of M = 10, a Lipschitz constant of L = 1, and t from zero to four. Using the equation given, we obtain the following: which means that h must be smaller than the above to get the desired error or less, and 4 / h, or about 1072 iterations will need to be completed to do so.
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