Fourier-splitting method for solving hyperbolic LQR problems

Figure 1.  Solution $w(t, \xi)$ to the one-dimensional advection equation (7) without control ($B = 0$) at time $t = 3.2$ by using Godunov's scheme with time step $\tau = 10^{-3}$

Figure 2.  Solution $w(t, \xi)$ to the one-dimensional advection equation (7) without control ($B = 0$) at time $t = 3.2$ by using Lax-Wendroff scheme (right panel) with time step $\tau = 10^{-3}$

Figure 3.  Volume ratio $\mathcal V_1(t)$ of the one-dimensional advection equation (7) without control ($B = 0$) by using Godunov's scheme or Lax-Wendroff scheme with time step $\tau = 10^{-3}$

Figure 4.  Solution $w(t, \xi)$ to the one-dimensional advection equation (7) with distributed control ($B = \text{I}_{\mathcal U}$) at time $t = 3.2$ by using Godunov's scheme with time step $\tau = 10^{-3}$

Figure 5.  Solution $w(t, \xi)$ to the one-dimensional advection equation (7) with distributed control ($B = \text{I}_{\mathcal U}$) at time $t = 3.2$ by using Lax-Wendroff scheme with time step $\tau = 10^{-3}$

Figure 6.  Volume ratio $\mathcal V_1(t)$ of the one-dimensional advection equation (7) with distributed control ($B = \text{I}_{\mathcal U}$) at time $t = 3.2$ by using Godunov's scheme or Lax-Wendroff scheme with time step $\tau = 10^{-3}$

Figure 7.  Solution $w(t, \xi)$ to the one-dimensional advection equation (7) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 3.2$ by using Godunov's scheme with time step $\tau = 10^{-3}$

Figure 8.  Solution $w(t, \xi)$ to the one-dimensional advection equation (7) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 3.2$ by using Lax-Wendroff scheme with time step $\tau = 10^{-3}$

Figure 9.  Volume ratio $\mathcal V_1(t)$ of the one-dimensional advection equation (7) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 3.2$ by using Godunov's scheme with time step $\tau = 10^{-3}$

Figure 10.  Volume ratio $\mathcal V_1(t)$ of the one-dimensional advection equation (7) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 3.2$ by using Lax-Wendroff scheme with time step $\tau = 10^{-3}$

Figure 11.  Enlargements of Figure 9

Figure 12.  Enlargements of Figure 10

Figure 13.  Solution to two-dimensional advection equation (8) at time $t = 3$ without control ($B = 0$) by using Fourier's method (left column) and Lax-Wendroff scheme (right column) with time step $\tau = 10^{-3}$ and number of grid points $N_\xi = N_\eta = 32, 64,128$ from top to bottom, respectively

Figure 14.  Solution to two-dimensional advection equation (8) at time $t = 3$ with distributed control ($B = \text{I}_{\mathcal U}$) by using Lax-Wendroff scheme (left column) and Fourier-Splitting (right column) with time step $\tau = 10^{-3}$ and number of grid points $N_\xi = N_\eta = 32, 64$ from top to bottom, respectively

Figure 15.  Volume ratio $\mathcal V_2(t)$ of two-dimensional advection equation (8) with distributed control ($B = \text{I}_{\mathcal U}$) by using Lax-Wendroff scheme and Fourier-Splitting with time step $\tau = 10^{-3}$

Figure 16.  Solution to the one-dimensional linearized shallow water equations (9) without control ($B = 0$) at time $t = 25$ with time step $\tau = 10^{-4}$

Figure 17.  Time-evolution of the volume ratio $\mathcal V_3$ for the one-dimensional linearized shallow water equations (9) without control ($B = 0$) at time $t = 25$ with time step $\tau = 10^{-4}$

Figure 18.  Solution to the one-dimensional linearized shallow water equations (9) with distributed control ($B = \text{I}_{\mathcal U}$) at time $t = 25$ with time step $\tau = 10^{-4}$

Figure 19.  Time-evolution of the volume ratio $\mathcal V_3$ for the one-dimensional linearized shallow water equations (9) with distributed control ($B = \text{I}_{\mathcal U}$) at time $t = 25$ with time step $\tau = 10^{-4}$

Figure 20.  Solution to the one-dimensional linearized shallow water equations (9) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 40$ with time step $\tau = 10^{-4}$

Figure 21.  Time-evolution of the volume ratio $\mathcal V_3$ for the one-dimensional linearized shallow water equations (9) with sink-like control ($B = \text{I}_{\Gamma_1}$) at time $t = 40$ with time step $\tau = 10^{-4}$

Figure 22.  Solution to two-dimensional shallow water equations (12) at time $t = 4.5$ without control ($B = 0$) by using Fourier's method (left column) and Lax-Wendroff scheme (right column) with time step $\tau = 10^{-4}$ and number of grid points $N_\xi = N_\eta = 16, 32, 64$ from top to bottom, respectively

Figure 23.  Solution to two-dimensional shallow water equations (12) at time $t = 4.5$ with distributed control ($B = \text{I}_{\mathcal U}$) by using Lax-Wendroff scheme (left column) and Fourier-Splitting (right column) with time step $\tau = 10^{-4}$ and number of grid points $N_\xi = N_\eta = 16, 32$ from top to bottom, respectively

Figure 24.  Volume ratio $\mathcal V_4$ for two-dimensional shallow water equations (12) for distributed control ($B = \text{I}_{\mathcal U}$) by using time step $\tau = 10^{-4}$

Figure 25.  Solution to two-dimensional shallow water equations (12) at time $t = 4.5$ with sink-like control ($B = \text{I}_{\Gamma_{2\ell}}$) by using Lax-Wendroff scheme (left column) and Fourier-Splitting (right column) with time step $\tau = 10^{-4}$ and number of grid points $N_\xi = N_\eta = 16, 32$ from top to bottom, respectively

Figure 26.  Volume ratio $\mathcal V_4$ for two-dimensional shallow water equations (12) for the sink-like control ($B = \text{I}_{\Gamma_1}$)

Figure 27.  Two-dimensional linearized shallow water equations with control matrix $B = \Gamma_{2r}$ representing a sink. Comparison between no control (left column), sequential splitting (column in the middle) and Strang splitting (right column)