Silent error detection in numerical time-stepping schemes

Errors due to hardware or low-level software problems, if detected, can be fixed by various schemes, such as recomputation from a checkpoint. Silent errors are errors in application state that have escaped low-level error detection. At extreme scale, where machines can perform astronomically many operations per second, silent errors threaten the validity of computed results. We propose a new paradigm for detecting silent errors at the application level. Our central idea is to frequently compare computed values to those provided by a cheap checking computation, and to build error detectors based on the difference between the two output sequences. Numerical analysis provides us with usable checking computations for the solution of initial-value problems in ODEs and PDEs, arguably the most common problems in computational science. Here, we provide, optimize, and test methods based on Runge–Kutta and linear multistep methods for ODEs, and on implicit and explicit finite difference schemes for PDEs. We take the heat equation and Navier–Stokes equations as examples. In tests with artificially injected errors, this approach effectively detects almost all meaningful errors, without significant slowdown.