BetaT

BetaT

Application

Naiver Strokes is an equation for Flow Velocity.

Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations they can be used to model and study magnetohydrodynamics. courtesy of wikipedia ("https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations")

problem

The problem with this application comes in the main function trying to calculate the finite-difference

The user inputs 2 values which will be used as a reference for the loop.

 // Finite-difference loop:
 for (int it=1; it<=nt-1; it++)
   {
     for (int k=0; k<=nx-1; k++)
   {
     un[k][it-1] = u[k][it-1];
   }
     for (int i=1; i<=nx-1; i++)
   {
     u[0][it] = un[1][it-1];
     u[i][it] = un[i][it-1] - c*dt/dx*(un[i][it-1]-un[i-1][it-1]);
   }
   }

Tests ran with no optimization on linux

Naiver Equation

n	Time in Milliseconds
100 x 100	24
500 x 500	352
1000 x 1000	1090
2000 x 2000	3936
5000 x 5000	37799
5000 x 10000	65955
10000 x 10000	118682
12500 x 12500	220198

gprof

it gets a bit messy down there, but basically 89.19% of the program is spent in the main() calculating those for loops shown above. The additional time is spent allocating the memory which might cause some slowdown when transferring it to the GPU across the bus int he future.

But the main thing to take away here is that main() is 89.19% and takes 97 seconds.

Each sample counts as 0.01 seconds.

 %   cumulative   self              self     total           
time   seconds   seconds    calls   s/call   s/call  name    
89.19     97.08    97.08                             main
 4.73    102.22     5.14 1406087506     0.00     0.00  std::vector<std::vector<double, std::allocator<double> >, std::allocator<std::vector<double, std::allocator<double> > > >::operator[](unsigned int)
 4.49    107.11     4.88 1406087506     0.00     0.00  std::vector<double, std::allocator<double> >::operator[](unsigned int)

Potential Speed Increase with Amdahls Law

Using Amdahls Law ---- > Sn = 1 / ( 1 - P + P/n )

We can examine how fast out program is capable of increasing its speed.

P = is the part of the program we want to optimize which from above is 89.17% n = the amount of processors we will use. One GPU card has 384 processors or CUDA cores and another GPU we will use has 1020 processor or CUDA cores.

Applying the algorithm gives us.

Amdahls Law for GPU with 384 Cores---- > Sn = 1 / ( 1 - 0.8919 + 0.8919/384 )

                                        Sn = 9.0561125222

Amdahls Law for GPU with 1024 Cores---- > Sn = 1 / ( 1 - 0.8919 + 0.8919/1024 )

                                         Sn = 9.176753777

Therefor According to Amdahls law we can expect a 9x increase in speed.

97 seconds to execute main / 9 amdahls law = 10.7777 seconds to execute after using GPU

Interestingly according to the law the difference in GPU cores does not significantly increase speed. Future tests will confirm or deny these results.

---

Potential Speed Increase with Gustafsons Law

Gustafsons Law S(n) = n - ( 1 - P ) ∙ ( n - 1 )

(Quadro K2000 GPU) S = 380 - ( 1 - .8918 ) * ( 380 - 1 ) = 339.031

(GeForce GTX960 GPU) S = 1024 - ( 1 - .8918 ) * ( 1024 - 1 ) = 913.3114

Using Gustafsons law we see drastic changes in the amount speed increase, this time the additional Cores made a big difference and applying these speed ups we get:

(Quadro K2000 GPU) 97 seconds to execute / 339.031 = 0.29

(GeForce GTX960 GPU) 97 seconds to execute / 913.3114 = 0.11

Tests ran with no optimization on Windows nisghts

System Specifications