A Geometric Approach for solving Heterogeneous Agent models: the Discrete Exterior Calculus scheme

By integrating Discrete Exterior Calculus (DEC) with upwind stabilization, I develop a structure-preserving numerical scheme that guarantees absolute mass conservation in heterogeneous agent models. Standard finite difference methods rely on algebraic approximations that entangle state dimensions. DEC, instead, natively isolates economic forces: exogenous income diffusion operates strictly on vertices, while endogenous savings advection flows across edges. This topological modularity rigorously justifies finite volume techniques on non-uniform grids and perfectly unifies the mechanism of different numerical methods used in continuous and discrete time, revealing the Endogenous Grid Method and Young’s projection as exact geometric Pullback and Pushforward operators. Crucially, DEC’s strict operator separation unlocks Strang splitting for transition dynamics. By safely decoupling income uncertainty from wealth accumulation, this fractional step method avoids dynamic multidimensional matrix inversions. Resolving advection via independent one-dimensional sweeps and pre-factorizing the static diffusion operator achieves more than a 2x computational speedup over purely finite difference iteration.