Abstract
In this paper, we answer the question under which conditions the porous-medium equation with convection and with periodic boundary conditions possesses gradient-type Lyapunov functionals (first-order entropies). It is shown that the weighted sum of first-order and zeroth-order entropies are Lyapunov functionals if the weight for the zeroth-order entropy is sufficiently large, depending on the strength of the convection. This provides new a priori estimates for the convective porous-medium equation. The proof is based on an extension of the algorithmic entropy construction method which is based on systematic integration by parts, formulated as a polynomial decision problem.
Highlights
IntroductionOur method below is related to a second order WKB-approximation (i.e. truncation of the Ansatz (1.3) after p = 2), which uses the refined asymptotics exp φ(x) ≈ C i ε x 0 φ(τ
This paper deals with an asymptotic scheme for the numerical solution of highly oscillating differential equations of the type ε2φ′′(x) + a(x)φ(x) = 0, (1.1)where 0 < ε ≪ 1 is a very small parameter and a(x) ≥ a0 > 0 a sufficiently smooth function
A fine mesh means high numerical costs, implying the inefficiency of standard methods. Such problems that require the numerical integration of highly oscillatory equations play an essential role in a wide range of physical phenomena: electromagnetic and acoustic scattering (Maxwell and Helmholtz equations in the high frequency regime), wave evolution problems in quantum and plasma physics (Schrodinger equation in the semiclassical regime), stiff mechanical systems, and so on
Summary
Our method below is related to a second order WKB-approximation (i.e. truncation of the Ansatz (1.3) after p = 2), which uses the refined asymptotics exp φ(x) ≈ C i ε x 0 φ(τ. Numerical errors of order O(hγ) in the phase-computation will typically induce O(hγ/ε) errors in the oscillatory integral. The asymptotic method [9, 18] provides approximations of arbitrarily high ε–order Since it does not yield high h–orders (cf §2.2 for details), it is not usable for constructing ODE schemes. The idea is to subtract from the oscillatory term exp of the integrand a trigonometric polynomial of the phase φ (which is appropriately compensated in the integration by parts) This procedure creates zeros in the integrand and increases the h–order of the error.
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