This paper presents the complete mathematical and numerical foundations of the Solar-Water-Ecosystem Digital Twin (SWEDT), a computational framework for assessing climate resilience in semi-arid regions. The governing equations are derived systematically from the three-dimensional Navier-Stokes equations through a hierarchy of physical assumptions, with each assumption rigorously justified via scaling analysis of the characteristic flow regime. A formal asymptotic analysis reveals that the hydrostatic approximation is valid to
for the study site, confirming the appropriateness of the shallow water equations. The numerical implementation employs a Godunov-type finite volume method with Roe’s approximate Riemann solver, MUSCL reconstruction for second-order spatial accuracy, and a TVD-preserving van Leer limiter. A novel Implicit-Explicit (IMEX) treatment of the Manning friction term is developed to handle stiffness at low depths, with stability analysis proving the scheme remains asymptotic-preserving as water depth approaches zero. The framework is validated for a 150 km2 catchment in Kitui County, Kenya, using open-access Earth Observation data: Integrated Multi-satellitE Retrievals for GPM (IMERG) rainfall, Sentinel-1 Synthetic Aperture Radar (SAR) flood extents, Shuttle Radar Topography Mission (SRTM) topography, and in-situ gauge records. Mesh convergence analysis identifies an optimal 10 m resolution, capturing 82% of SAR-observed flood extent (F1 = 0.82), which compares favorably with recent studies using in-situ instrumentation. Truncation error analysis demonstrates that the leading-order error term scales as
, explaining the model’s sensitivity to bathymetric gradients. Scenario analyses quantify the potential of vegetative buffer strips (22% - 28% peak discharge reduction, physically attributed to increased friction length scales) and photovoltaic-battery systems (23% improvement in water supply
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