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# Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations

• * Corresponding author: Ansgar Jüngel
The authors acknowledge partial support from the Austrian Science Fund (FWF), grants F65, P30000, P33010, and W1245
• Structure-preserving finite-difference schemes for general nonlinear fourth-order parabolic equations on the one-dimensional torus are derived. Examples include the thin-film and the Derrida–Lebowitz–Speer–Spohn equations. The schemes conserve the mass and dissipate the entropy. The scheme associated to the logarithmic entropy also preserves the positivity. The idea of the derivation is to reformulate the equations in such a way that the chain rule is avoided. A central finite-difference discretization is then applied to the reformulation. In this way, the same dissipation rates as in the continuous case are recovered. The strategy can be extended to a multi-dimensional thin-film equation. Numerical examples in one and two space dimensions illustrate the dissipation properties.

Mathematics Subject Classification: Primary: 35K30; Secondary: 35Q68, 65M06, 65M12.

 Citation: • • Figure 1.  Evolution of the DLSS equation in a semi-logarithmic scale, using the initial datum $u^0(x) = \max\{10^{-10}, \cos(\pi x)^{16}\}$

Figure 2.  Left: Decay of the logarithmic entropy $s_0(u(t))$ for two different space grid sizes $h = 1/20$ and $h = 1/200$. Right: Convergence of the $\ell^2$ error. The dots are the values from the numerical solution, the solid line is the regression curve

Figure 3.  Left: Decay of the Shannon entropy $s_1(u(t))$ with $h = 1/100$. Right: Convergence of the $\ell^2$ error. The dots are the values from the numerical solution, the solid line is the regression curve

Figure 4.  Evolution of the solution to the thin-film equation at times $t = 0$ (densely dotted), $t = 2\cdot 10^{-4}$ (dotted), $t = 5\cdot 10^{-4}$ (dash-dotted), $t = 1\cdot 10^{-3}$ (dashed), $t = 2\cdot 10^{-3}$ (densely dashed), and $t = 5\cdot 10^{-3}$ (solid) and grid sizes $h = 1/10$ (left), $h = 1/200$ (right)

Figure 5.  Decay of the logarithmic entropy $S_0(u(t))$ for various space grid sizes

Figure 6.  Evolution of the solution to the two-dimensional thin-film equation with $\beta = 2$, $t = 0$ (top left), $t = 3\cdot 10^{-9}$ (top right), $t = 10^{-8}$ (bottom left), $t = 10^{-6}$ (bottom right)

Figure 7.  Decay of the logarithmic entropy $S_0(u(t))$ for various space grid sizes

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