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3.5 Results 79
in the L2 cache of the processor comfortably. However the architecture implemented
on the FPGA was not optimized for speed and runs only on 100MHz,
its performance is comparable or in some cases slightly higher than a 3GHz Intel
Core 2 Duo microprocessor.
3.5 Results
The L ∞ norm of the error in case of different grid resolution and different fixedpoint
and floating-point precisions are compared in Figure (3.6) – (3.8). As the
resolution is increased, the accuracy of the solution can be improved. Finer
grid resolution requires smaller time-step, according to the CFL condition, which
results in more operations. After a certain number of steps the round-off error
of the computation will be comparable or larger than the truncation error of the
numerical method. Therefore the step size can not be decreased to zero in a
given precision. The precision of the arithmetic units should be increased when
the grid is refined.
The arithmetic unit in the 2nd order case (see Figure 3.7) is twice as large as
in 1st order arithmetic unit, additionally the step-size should be halved therefore
four times more computation is required. The slope of the error is higher as
shown in Figure 3.9, therefore computation on a coarser grid results in higher
accuracy.
The L ∞ norm of fix-point and floating point solutions with the same mantissa
width are compared in Figure 3.9. Because we use 11 bit for the exponent in
floating-point numbers the 40 bit floating point number compared to the 29 bit
fix-point number. As it can be seen, the floating-point computation results in
more accurate solution than the fix-point. But just adding four bits to the 29bit
fixed-point number and using 33bit fixed-point number the L ∞ norm of the error
is comparable to the error of the 40bit floating point computations. In addition
7 bits are saved, which results in lower memory bandwidth and has a reduced
area requirement. The fixed-point arithmetic unit requires 22 multipliers while
the floating-point arithmetic unit requires only 16 multipliers, the number of
required slices is far more less in the fixed-point case. While the 34 bit fixed-point