Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations

Figure 2.  Evolution of a discrete solution $u_\Delta$, evaluated at different times $t = 0,0.05,0.1,0.15,0.175,0.25$ (from top left to bottom right). The red line is the corresponding Barenblatt-profile ${{\text{b}}_{\alpha ,\lambda }}$.

Figure 1.  Left: Numerically observed decay of $H_{\alpha,\lambda}(t)-{\mathbf{H}_{\alpha ,\lambda }^{\min }}$ and $F_{\alpha,\lambda}(t)-{\mathbf{F}_{\alpha ,\lambda }^{\min }}$ along a time period of $t\in[0,0.8]$, using $K=25,50,100,200$, in comparison to the upper bounds $({\mathcal{H}_{\alpha ,\lambda }}(u^0)-{\mathcal{H}_{\alpha ,\lambda }}({{\text{b}}_{\alpha ,\lambda }}))\exp(-2\lambda t)$ and $({\mathcal{F}_{\alpha ,\lambda }}(u^0)-{\mathcal{F}_{\alpha ,\lambda }}({{\text{b}}_{\alpha ,\lambda }}))\exp(-2\lambda t)$, respectively. Right: Convergence of discrete minimizers $u_{\delta }^{\min }$ with a rate of $K^{-1.5}$.

Figure 3.  Snapshots of the densities $\text{b}_{\alpha ,0}^{*}(t,\cdot)$ (red lines) and $u_\Delta$ (black lines) for the initial condition $\text{b}_{\alpha ,0}^{*}(0,\cdot)$ at times $t=0$ and $t= 0.1\cdot 10^{i}$, $i=0,\ldots,3$, using $K=50$ grid points and the time step size $\tau=10^{-3}$.