Linear Advection Problem
Discontinuous Galerkin Spectral Element Method
Nodal basis functions on a Gauss-Lobatto grid
Numerical quadrature
Upwind numerical flux
Third order explicit SSP-RK
15 Cubic elements
I.C: max(0, u(0, x) = sin(x))...

From rickardbj

Real-time audio spectrum visualization
8192-point FFT and 400 log-equal bands
C++, OpenGL, WASAPI
Song is Slumberjack by Raina (Milkytracker)...

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Navier-Stokes problem in a lid driven cavity (Re = 250).
Semi-implicit finite difference approximation using Chorin-Temam projection on a staggered grid. Direct solvers for the diffusion, preconditioned conjugate gradient for the pressure....

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Singular Value Decomposition
In the first step step, the matrix is reduced to bidiagonal form using a series of Householder reflections from left and right.
The diagonalization in the second step is performed iteratively using Givens rotations....

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One of the Arenstorf orbits, a periodic solution to the restricted three-body problem. Solved using DOPRI5.
Pay attention to the accumulation of errors!
More details in:
Hairer, Nørsett, Wanner: Solving ordinary differential equations I...

From rickardbj