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

#### 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 $e^{C\ell^{-1/2} }$.
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Colombini, Ferruccio
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/417667
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