Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs

Figure 1.  Different filters

Figure 2.  Different basis of orthogonal polynomials: Chebyshev polynomials (top left), Legendre polynomials (top right) and Fourier polynomials (bottom)

Figure 3.  Hierarchy of the triangulation of $ \Omega = ]0,1[^2 $: solid line (coarse triangulation), dashed line (complementary triangulation)

Figure 4.  3D output and iso-values for $ u(x,y) = \cos \bigl(5(1-x^2-y^2)\bigr) $ on the unit disk. The function $ u $ with $ \mathbb{P}_2 $ elements (top) and the $ z_h $ components (bottom)

Figure 5.  Function $ u(x,y) = \cos \bigl(5(1-x^2-y^2)\bigr) $ on the unit disk. Eigenfunction (Fourier) coefficients on the fine mesh for the original function $ u_h $ (red line) and the associated correction $ z_h $ (blue line)

Figure 6.  Decomposition of the signal $ u(x) = \sin(2\pi x)+\sin(6\pi x)+\sin(12\pi x)+0.1\sin(20\pi x)+0.1\sin(30\pi x)+0.1\sin(120\pi x) $, $ N = 100, m = 8 $. Original signal (top), high frequencies fluctuent part (bottom left) and low frequencies mean part (bottom right)

Figure 7.  Dirichlet BC (Top Left), Neumannn BC (Top Right), Periodic BC (Bottom). The original values are given at grid points $ \times $, interpolated values are computed at grid points $ o $. The boundary points are marked by a dot, in the Dirichlet case

Figure 8.  Low-pass filters: (left) - Exponential of low pass filters = high-pass filter (right)

Figure 9.  Coarse mesh (left) and fine mesh (right)

Figure 10.  Initial solution (left) and Final Solution (right) - \hskip 1.cm $ \mathbb{P}_2 $ Elements. $ \epsilon=0.08 $, $ \Delta t =1.e-4, \tau=4\epsilon^2, T=0.012 $

Figure 11.  Energy (left) and mass (right) vs time - \hskip 2.cm $ \mathbb{P}_2 $ Elements. $ \epsilon=0.08 $, $ \Delta t =1.e-4, \tau=4\epsilon^2, \ T=0.012 $

Figure 12.  $ \|u\|_{L^2} $ vs time with $ u_0 = \cos(2\pi x/L)+\cos(6\pi x/L)+\cos(12\pi x/L)+\cos(20\pi x/L) $, $ (\tau_0,\tau_1) = (10,100) $ (left) $ (\tau_0,\tau_1) = (100,10) $ (right), $ \Delta t = 1.e-1 $, $ T = 10 $, $ L = 100 $

Figure 13.  KdV $ (\tau_0,\tau_1) = (0,0) $, $ \mathbb{P}_1 $ Elements. $ \Delta t = 1.e-2 $, $ T = 40 $. Mass $ \int_0^Ludx $ (left) and $ L^2 $-norm $ |u|_{L^2} $ (right) vs time

Figure 14.  KdV $ (\tau_0,\tau_1) = (0,100) $ Low pass damping, $ \mathbb{P}_1 $ Elements. $ \Delta t = 1.e-2 $, $ T = 40 $. Mass $ \int_0^Ludx $ (left) and $ L^2 $-norm $ |u|_{L^2} $ (right) vs time

Figure 15.  KdV $ (\tau_0,\tau_1) = (0,100) $ Low pass damping, $ \mathbb{P}_1 $ Elements. $ \Delta t = 1.e-2 $, $ T = 4 $. Mass $ \int_0^Ludx $ (left) and $ L^2 $-norm $ |u|_{L^2} $ (right) vs time

Figure 16.  Heat Equation - $ L^{\infty} $-norm of the error vs time - $ n = 100, m = 4, \Delta t = 9.95 \, 10^{-5}, \tau = 1.6,10^{4} $

Figure 17.  KSE Low and high frequency components of the solution at final time $ T=140 $ (left), time evolution of the mean value (right) - $ n=128, \ m=8 $, $ \Delta t=0.01 $, $ L=10 $ (line 1), $ L=20 $ (line 2)

Figure 18.  KSE Low and high frequency components of the solution at final time $ T=140 $ (left), time evolution of the mean value (right) - $ \Delta t=0.01 $. Line 1: $ L=50 $, $ n=128, \ m=8 $; line 2: $ L=100 $, $ n=256, \, m=10 $