advection_pde, an Octave code which solves the advection PDE dudt + c * dudx = 0 in one spatial dimension and time, with a constant velocity c, and periodic boundary conditions, using the FTCS method, forward time difference, centered space difference.
We solve for u(x,t), the solution of the constant-velocity advection equation in 1D,
du/dt + c du/dx = 0
over the interval:
0.0 <= x <= 1.0
with constant velocity:
c = 1
with periodic boundary conditions:
u(0,t) = u(1,t) for all t
and initial condition
u(x,0) = (10x-6)^2 (8-10x)^2 for 0.6 <= x <= 0.8
= 0 elsewhere.
We use a method known as FTCS (forward time, centered space):
The advection velocity is constant in time and space. Therefore, (given our periodic boundary conditions), the solution should simply move smoothly from left to right, returning on the left again. However, the FTCS numerical method is unstable for this advection problem. Hence, the numerical solution will include erroneous oscillations which spread and blow up over time. There are more sophisticated methods for the advection problem, which do not exhibit this instability.
The computer code and data files described and made available on this web page are distributed under the MIT license
advection_pde is available in a MATLAB version and an Octave version.
allen_cahn_pde, an Octave code which sets up and solves the Allen-Cahn reaction-diffusion system of partial differential equations (PDE) in 1 space dimension and time.
artery_pde. an Octave code which solves a partial differential equation (PDE) that models the displacement of arterial walls under pressure.
diffusion_pde, an Octave code which solves the diffusion partial differential equation (PDE) dudt - mu * d2udx2 = 0 in one spatial dimension, with a constant diffusion coefficient mu, and periodic boundary conditions, using FTCS, the forward time difference, centered space difference method.
gray_scott_pde, an Octave code which solves the partial differential equation (PDE) known as the Gray-Scott reaction diffusion equation, displaying a sequence of solutions as time progresses.
schroedinger_linear_pde, an Octave code which solves the complex partial differential equation (PDE) known as Schroedinger's linear equation: dudt = i uxx, in one spatial dimension, with Neumann boundary conditions.
schroedinger_nonlinear_pde, an Octave code which solves the complex partial differential equation (PDE) known as Schroedinger's nonlinear equation: dudt = i uxx + gamma * |u|^2 u, in one spatial dimension, with Neumann boundary conditions.
spiral_pde, an Octave code which solves a pair of reaction-diffusion partial differential equations (PDE) over a rectangular domain with periodic boundary condition, whose solution is known to evolve into a pair of spirals.
string_pde, an Octave code which sets up and solves the partial differential equations (PDE) describing a vibrating string.
tumor_pde, an Octave code which solves the tumor angiogenesis partial differential equations (PDE) using MATLAB's pdepe() function.
wave_pde, an Octave code which uses finite differences in space, and the method of lines in time, to set up and solve the partial differential equations (PDE) known as the wave equations, utt = c uxx.