Spectral-diagonalization-based matrix exponential integration for efficient and stable solutions of full-Bloch equations in surface NMR - Lin, Tingting; Wang, Qingyue; Jiang, Chuandong; Ren, Chunpeng; Wang, Yunzhi; Wang, Liang (Computers & geosciences, 2026)
Surface nuclear magnetic resonance (SNMR) is a geophysical extension of nuclear magnetic resonance (NMR) that enables non-invasive mapping of subsurface hydrogeological properties by measuring the relaxation response of groundwater hydrogen nuclei. Accurately modeling the transient spin dynamics in SNMR requires solving the full-Bloch equations under Earth’s geomagnetic field, where magnetic field inhomogeneities, multicomponent relaxation, and nonlinear pulsed excitations introduce significant mathematical and computational challenges. We present a spectral-diagonalization-based matrix exponential integration (SD-MEI) algorithm for efficient and stable solutions of full-Bloch equations in SNMR. Conventional explicit numerical methods exhibit cumulative discretization errors and escalating computational costs due to step-size dependence and finite precision limitations. SD-MEI integrates spectral diagonalization with matrix exponential operations, replacing iterative computations with a single eigendecomposition of the system matrix. This approach achieves parameter-robust computational complexity while maintaining numerical stability across broad B1 field strengths (10−10 T to 10−5 T) and relaxation times (10 ms to 1000 ms). Validated for steady-state free precession (SSFP) dynamics in heterogeneous geomagnetic environments, the method enables high-accuracy modeling of transient magnetization evolution with large time steps. The framework advances SNMR efficient forward modeling and inversion while optimizing protocols by resolving critical limitations in existing numerical and analytical approaches. •Modeling of Magnetization Dynamics: Accurately describes magnetization vectors over broad excitation fields.•Control of Computational Complexity: Keeps computation stable, improving efficiency by 20–1000x.•Improved Numerical Stability: Solves convergence issues in stiff and highly nonlinear systems.•Temporal Flexibility: Computes magnetization at any time without affecting overall accuracy.