Adaptive Spectral Block Floating Point for Discontinuous Galerkin Methods

Discontinuous Galerkin (DG) methods are powerful tools for discretizing partial differential equations, but their application to large-scale problems is often constrained by significant memory demands. We introduce a Spectral Block Floating Point (SBFP) data type specifically designed to exploit the spectrally decaying polynomial coefficients of modal DG discretizations, whose higher-order coefficients progressively diminish in magnitude if the solution is smooth. For an 1D problem, SBFP achieves a remarkable 33% reduction in memory overhead compared to standard full-precision storage – by packing the shared exponent of the constant term along with truncated mantissas of the linear and quadratic coefficients of a quadratic DG solution into a single 64-bit word. To reduce the potential losses of significant mantissa bits due to large variations in exponents, we provide an adaptive variant of the SFBP data type, termed Adaptive Spectral Block Floating Point (ASBFP). ASBFP introduces "specialization bits" that facilitate unique configurations, encompassing a base shared-exponent mode and many biased configurations for the linear and quadratic terms. This adaptive approach improves the way mantissas align across coefficients, helping to reduce precision loss. Numerical experiments show that ASBFP successfully maintains the accuracy expected from standard floating-point formats while still delivering notable savings in memory usage.