The Lax-Friedrichs scheme for interaction between the inviscid Burgers equation and multiple particles

Figure 4.  Example 8.3. Particle trajectories using basic scheme (upper plot) and MUSCL (lower plot). Both the true (thick line) and approximate (thin line) trajectories are plotted. For the MUSCL scheme (lower plot) the true and approximate trajectories are visually indistinguishable at this level of discretization. $ \Delta x = 1.95 \times 10^{-5} $, $ \mu = .25 $, 102401 time steps

Figure 5.  Example 8.4. Basic scheme (left) and MUSCL (right). The horizontal axis represents $ x $, and the vertical axis represents $ t $. Top level plots: $ \Delta x_1 = 3.75 \times 10^{-4} $. Middle level plots: $ \Delta x_2 = {1\over 2} \Delta x_1 $. Bottom level plots: $ \Delta x_3 = {1\over 4} \Delta x_1 $. $ \mu = .125 $ for all plots

Figure 1.  Example 8.1. Top: Fluid velocity $ u $ at $ t = 1 $. Exact solution is solid line, with sharp corners. Bottom: Particle position error vs. time. Basic scheme (left plots) and MUSCL scheme (right plots). $ \Delta x = .0025 $ (dashed line), and $ \Delta x = .00125 $ (solid line). Both approximations used $ \mu = .25 $

Figure 2.  Example 8.2. Fluid velocity $ u $ at $ t = 1 $. Basic scheme (left plots) and MUSCL scheme (right plots). Exact solution (dashed line) and approximate solution (solid line). Top plots used $ \Delta x = .005 $, bottom plots used $ \Delta x = .000625 $. All approximations used $ \mu = .25 $. A spurious kink is visible. Its magnitude diminishes with grid refinement

Figure 3.  Example 8.3. Solution $ u $ using basic scheme at $ t = .125 $ (upper left), and using MUSCL (upper right). True solution (dashed line) and approximate solution (solid line). Both upper plots computed with $ \Delta x = .00325 $, $ \mu = .25 $. The lower plots show the error in $ u $ in discrete $ L^1 $ norm as a function of time using the basic scheme (lower left) and MUSCL scheme (lower right). Uses $ \Delta x = .00325 $ and $ \Delta x = .001625 $, $ \mu = .25 $