PPKE ITK PhD and MPhil Thesis Classes

72
3. INVESTIGATING THE PRECISION OF PDE SOLVER
ARCHITECTURES ON FPGAS
3.2.2 The Second-order Limited Scheme
The overall accuracy of the scheme can be raised to second-order if the spatial and
the temporal derivatives are calculated by a second-order approximation. One
way to satisfy the latter requirement is to perform a piecewise linear extrapolation
of the primitive variables P L and P R at the two sides of the interface in (3.5).
This procedure requires the introduction of additional cells with respect to the
interface, i.e. cell LL (left to cell L) and cell RR (right to cell R). With these
labels the reconstructed primitive variables are
with
P L = P L + g L (δP L , δP C )
2
, P R = P R − g R (δP C , δP R )
, (3.8)
2
δP L = P L − P LL , δP C = P R − P L , δP R = P RR − P R (3.9)
while g L and g R are the limiter functions.
The previous scheme yields acceptable second-order time-accurate approximation
of the solution, only if the variations in the flow field are smooth. In case
of discontinuous initial conditions this simple second order approximation can
not be used, because the method can be unstable. In order to capture these discontinuities
without spurious oscillations, in (3.8) we apply the minmod limiter
function, also:
⎧
⎪⎨
g L (δP L , δP C ) =
⎪⎩
δP L if |δP L | < |δP C |
and δP L δP C > 0
δP C if |δP C | < |δP L |
and δP L δP C > 0
0 if δP L δP C ≤ 0
The function g R (δP C , δP R ) can be defined analogously.
(3.10)
3.3 Testing Methodology
The goal of the experiment was to assign an adequate step size and an optimal
precision for the problem, which is described by equation (3.1). During the
solution different fixed-point and floating point numbers are used. The methodology
of the experiment is the investigation of different precision on different grid
resolution.