pce_burgers, a MATLAB code which defines and solves a version of the time-dependent viscous Burgers equation, with uncertain viscosity, using a polynomial chaos expansion, in terms of Hermite polynomials, by Gianluca Iaccarino.
The time-dependent viscous Burgers equation to be solved is:
du/dt = - d ( u*(1/2-u)) /dx + nu d2u/dx2 for -3.0 <= x <= 3.0
with boundary conditions
u(-3.0) = 0.0, u(+3.0) = 1.0.
The viscosity nu is assumed to be an uncertain quantity with normal distribution of known mean and variance.
A polynomial chaos expansion is to be used, with Hermite polynomial basis functions h(i,x), 0 <= i <= n.
Because the first two Hermite polynomials are simply 1 and x, we have that
nu = nu_mean * h(0,x) + nu_variance * h(1,x).
We replace the time derivative by an explicit Euler approximation,
so that the equation now describes the value of U(x,t+dt) in terms
of known data at time t.
Now assume that the solution U(x,t) can be approximated by the truncated expansion:
U(x,t) = sum ( 0 <= i <= n ) c(i,t) * h(i,x)
In the equation, we replace U by its expansion, and then multiply
successively by each of the basis functions h(*,x) to get a set of
n+1 equations that can be used to determine the values of c(i,t+dt).
This process is repeated until the desired final time is reached.
At any time, the coefficients c(0,t) contain information definining the expected value of u(x,t) at that time, while the higher order coefficients can be used to determine higher moments.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
pce_burgers is available in a C version and a C++ version and a Fortran90 version and a MATLAB version.
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The original FORTRAN90 version by Gianluca Iaccarino. MATLAB version by John Burkardt.