This paper analyses the approximate solution of very weakly hyperbolic Cauchy problems. These problems have very sensitive dependence on initial data. We treat a single family of such problems showing that in spite of the sensitive dependence, approximate solutions with desired precision $\eps$ can be computed in finite precision arithmetic with cost growing polynomially in $1/\eps$. The sensitive dependence requires high finite precision. The analysis required a new Gevrey stability estimate for the leap frog scheme. The latter depends on a new discrete Glaeser inequality. The cost of calculating solutions with features on scale $\ell \ll 1$ grows as $e^{C\ell^{-1/2} }$.

Numerical Analysis of Very Weakly Well Posed Hyperbolic Cauchy Problems

COLOMBINI, FERRUCCIO
2015

Abstract

This paper analyses the approximate solution of very weakly hyperbolic Cauchy problems. These problems have very sensitive dependence on initial data. We treat a single family of such problems showing that in spite of the sensitive dependence, approximate solutions with desired precision $\eps$ can be computed in finite precision arithmetic with cost growing polynomially in $1/\eps$. The sensitive dependence requires high finite precision. The analysis required a new Gevrey stability estimate for the leap frog scheme. The latter depends on a new discrete Glaeser inequality. The cost of calculating solutions with features on scale $\ell \ll 1$ grows as $e^{C\ell^{-1/2} }$.
Colombini, Ferruccio
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11568/417667
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