Figures from top to bottom

• The interface indicated by the magenta lines are going to split into top and bottom parts. The levelset equation is solved using a high order finite difference method.
• Acoustic waves are scattered at the sharp velocity discontinuity in inhomogeneous media. The two-way wave equation is discretized with the 4th order spatial and 2nd order temporal accuracy.
• Wall shear stress distribution on the blood vessel walls in the brain. The Navier-Stokes equations are solved via Spectral element emthod.
• The different behavior of two time marching schemes: leapfrog and trapezoidal. Both are symplectic and the phase diagram stay around the circle. However the implicit trapezoidal is superior to the explicit leapfrog.

Massachusetts Institute of Technology

Department of Mathematics