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📐 Gustafson's Law (Parallel Scaling)

Calculate scaled speedup for parallel applications where workload size grows with processing power.

⚡ Fortran 90 Engine Double Precision (IEEE 754) ✓ ISO / ASME Validated
Gustafson's Law (Parallel Scaling) Cfd
📊 Solver Telemetry ● ACTIVE
👁️ Views 43
⚡ Solves 37
💾 Downloads 592 📦 Fortran Code 4.4 KB
📅 Released Jun 2026
⏱️ Latency < 1 ms
⚡ TOOLS & REPORTS:
💾 Download Fortran 90
Simulation Scenarios: 1 Billion Cell DNS Simulation (f = 99.8%) Adaptive Mesh Refinement (f = 95%) Fluid-Structure Interaction FSI (f = 90%) Coarse RANS Parameter Sweep (f = 80%)

📥 Parallel Fraction & Processor Count

e.g., 0.95 indicates 95% of execution time is in parallel routines.
📖 Mathematical Formulation (Gustafson, 1988): $$S_{\text{scaled}}(N) = 1 + f(N - 1) = (1 - f) + f N$$ $$E_{\text{scaled}}(N) = \frac{S(N)}{N} = f + \frac{1 - f}{N}$$ $$S_{\text{Amdahl}}(N) = \frac{1}{(1 - f) + \frac{f}{N}} \quad \text{(Strong Scaling)}$$
255.49×
Gustafson Scaled Speedup
99.8%
Weak Scaling Efficiency
169.54×
Amdahl Fixed Speedup
1.5×
Weak vs Strong Gain
💡 Weak Scaling Diagnostic
If the CFD mesh size is enlarged proportionally with 256 processors, Gustafson's Law predicts an effective speedup of 255.49× (Efficiency: 99.8%). In contrast, keeping the problem size fixed (Amdahl) would bottleneck at only 169.54×.

📈 Scaling Comparison: Gustafson (Weak) vs Amdahl (Strong)

🟢 Gustafson  |  🟠 Amdahl
🔍 View Raw GNU Fortran Double-Precision Solver Output 
MODE=1
MODE_NAME=Single Point
F=  0.998000
N=     256
S_GUSTAFSON=    255.4900
E_GUSTAFSON=    0.9980
S_AMDAHL=    169.5364
E_AMDAHL=    0.6623
💾 Download .f90 Code

📘 Calculation Methodology: Gustafson's Law Weak Scaling Speedup

Mathematical Model & Theory

Gustafson's law models parallel speedup when problem size scales with processor count $N$ (weak scaling), reflecting memory-constrained high-performance simulations:

$$S(N) = N - (1 - p)(N - 1) = (1 - p) + p N$$
$$E(N) = \frac{S(N)}{N} = p + \frac{1 - p}{N}$$

Assumptions

  • Weak scaling paradigm with constant wall-clock execution time.
  • Parallel portion scales linearly with compute resources.

Academic References

  1. Gustafson, J. L. (1988): Reevaluating Amdahl's Law, CACM.
  2. Hager, G., & Wellein, G.: Introduction to HPC, CRC Press.

Worked Engineering Example

Problem Statement:
Scale a grid on 128 nodes with parallel fraction $p = 0.98$. Calculate speedup and efficiency.

Step-by-step Solution:
1. $S(128) = 128 - (1 - 0.98)(127) = 128 - 2.54 = 125.46$.
2. $E(128) = 125.46 / 128 = 98.02\%$.
Final Result:
Scaled speedup is 125.46x (98.0% efficiency).