# How error scales with numerical precision in molecular dynamics?

In terms of time-step, numerical error in molecular dynamics scales with square, i.e. $$error \approx dt^2$$. But how it look for numerical precision ? E.g. how much bigger will be numerical error when I use half precision, instead single and single instead double ?

• The method is Verlet? Which variant, velocity, leapfrog, something else? Did you de-singularize the potential? Is the step size small enough to have a sufficient sampling density during a collision or fly-by? Why would you prefer single or half precision? Is double precision readily available? May 16, 2021 at 10:24
• Which variant, velocity, leapfrog, something else - Leapfrog with BAOAB integration scheme. "Why would you prefer single or half precision" - speed. Nevertheless, more general answer would be preferable. May 16, 2021 at 10:45
• Before you consider speed can you answer how many significant figures you actually need to do the science you want to do? May 16, 2021 at 12:02

3. the error from accumulating the single updates of size $$O(h)$$ to the integration result of size $$O(1)$$.
So the total error in each step is something like $$C_1h^{p+1}+C_2\mu_{eval}h+C_2\mu_{acc}$$ where $$\mu_{eval}$$ is the machine constant for the number type used for evaluation and $$\mu_{acc}$$ the one for the number type used in the accumulation of the steps. The global error, in a first approach, results from multiplying with $$N=T/h$$, giving $$C_1h^pT+C_2\mu_{eval}T+C_2\frac{\mu_{acc}T}{h}$$ So ideally (in an academic situation where variables and derivatives have a scale close to $$1$$) you would like to have $$h^p\ge\mu_{eval}\ge\frac{\mu_{acc}}{h}$$. For $$p=2$$ using single precision in evaluation, $$\mu_{eval}=10^{-8}$$, and double precision for the accumulation, $$\mu_{acc}=10^{-15}$$, this would give errors according to the second order of the method down to $$h=10^{-4}$$. Another variant to achieve increased precision in the accumulation without a different data type is Kahan or compensated summation.
• An alternative to IEEE-754 double-precision (binary64) accumulation seems to be Q24.40 fixed-point accumulation as used by the GPU version of AMBER: Scott Le Grand, Andreas W. Götz, Ross C. Walker, "SPFP: Speed without compromise—A mixed precision model for GPU accelerated molecular dynamics simulations", Computer Physics Communications 184 (2013) 374–380 May 16, 2021 at 21:24