ISSN:
1432-2250
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
,
Physics
Notes:
Abstract We characterize ill-posed problems as catastrophically (Hadamard) unstable to short waves. The growth rate tends to infinity as the wavelength tends to zero. The mathematical description of ill-posed problems is framed in terms of instability. These problems cannot be integrated numerically; the finer the mesh, the worse is the result. The instability must be regularized. Ill-posed problems which arise in problems involving interfaces, oil recovery, granular media, and viscoelastic fluids are regularized in different ways, by adding effects of surface tension or viscosity or compressibility or by weakening the initial discontinuity. Problems which are stables t → ∞ for any fixed wavelength λ, no matter how small, can be Hadamard unstable with catastrophic instability as λ → 0 for a fixed t, no matter how large. We stress the utility of freezing coefficients in nonlinear and quasilinear systems and prove that in general ill-posed problems cannot be solved unless the initial data is analytic. We show why the shock up of first-order systems which are nonlinear in first derivatives can be expected to lead to discontinuities in second, rather than first, derivatives.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00418002
