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I think he has a little of that Al Gore stiff problem".
Numerical results in Tables 4, 5, 6, and 7 indicate that, even for a nonlinear equation, the computational accuracy of the IM method is improved for both a non-stiff problem and a stiff problem.
To accelerate the numerical integration, a "longitudinal constraint" is suggested to alleviate the stiff problem of the dynamic equations.
Table 2 highlights that, for the stiff problem, the Milstein method works unreliably and has large errors for not too small step sizes, whereas the IM method works very well.
In order to verify and validate the scheme, we applied the scheme to a stiff problem of the shock wave structure for all Mach numbers and to the two-dimensional hypersonic rarefied and low-speed microscale gas flows past a circular cylinder.
The full system of coupled nonlinear ODEs described above was solved as an initial value problem to steady state using MATLAB's 'ode15s' solver routine (a multi-step variable-order stiff problem solver that employs the numerical differentiation formulas scheme), with the relative error tolerance set at 10−6 (0.0001% accuracy) and an initial step size of 10−4.
Similar(54)
Similarly to Rosenbrock methods, the proposed scheme is well suited for stiff problems with the stiffness itself appearing in linear terms from the drift coefficients, and may lose efficiency in the case of stiff nonlinear terms.
This algorithm is specifically designed to solve stiff problems, and its performance increases with stiffness.
For strongly or infinitely stiff problems, there can occur an order reduction phenomenae.
The scheme is designed to be A-stable for highly stiff problems.
It maintains A-stability, which makes it appropriate for the solution of stiff problems.
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