MADNESS  0.10.1
Spectral propagator in time using semigroup approach

The source is here.

Points of interest
  • high-order integration of general time-dependent problems
  • semigroup approach
Background

Given $ u(0) $, we seek to solve the PDE

\[ \frac{du}{dt} = \hat{L} u + N(u) \]

for the function at some future time $ t $ (i.e., for $ u(t) $). $ \hat{L} $ is a linear operator that we are able to exponentiate, and $ N $ is everything else including linear and non-linear parts.

In the semigroup approach the formal solution to the PDE is written

\[ u(t) = \exp(t \hat{L} ) u(0) + \int_0^t \exp((t-\tau)\hat{L}) N(u(\tau)) d\tau \]

Numerical quadrature of the integral using Gauss-Legendre quadrature points is used resulting in a set of equations that are iteratively solved (presently using simple fixed point iteration from a first-order explicit rule).

The user provides

Have a look in testspectralprop.cc for example use.

With $ n $ quadrature points, the error is $ O\left(t^{2n+1}\right) $ and the number of applications of the exponential operator per time step is $ 1+(n_{it}+1)n +n_{it}n^2 $ where $ n_{it} $ is the number of iterations necessary to solve the equations (typically about 5 but this is problem dependent).