The Pratyaksh Framework: A Self-Limiting TVD Explicit Runge-Kutta Family for Real-Time Physics, Robotics, and Scientific Computing

Explicit Runge-Kutta methods, most notably the classical fourth-order scheme (RK4), remain the computational foundation of dynamical simulation across robotics, computer graphics, and machine learning due to their high accuracy and matrix-free evaluation. However, explicit polynomial integrators suffer from catastrophic numerical divergence (NaN overflow) whenever localized physical transients, contact impulses, or steep shock gradients violate linear stability boundaries. Conversely, unconditionally stable implicit integrators require constructing and inverting massive Jacobian matrices, rendering them computationally prohibitive for real-time deadlines or massively parallel GPU architectures.

In this paper, we introduce The Pratyaksh Framework, an explicit, matrix-free rational Runge-Kutta framework that dynamically bounds stage divergence through a dimensionless vector curvature ratio. By evaluating the denominator via a global inner-product norm, we resolve the classical coordinate-singularity pathology that historically plagued rational integrators. We derive two canonical formulations: Formula A (Order 2), which introduces non-linear viscosity providing strict Total Variation Diminishing (TVD) shock-capturing, and Formula B (Order 4), which preserves asymptotic fourth-order convergence down to machine precision. The Pratyaksh framework provides an explicit, matrix-free shock-absorbing alternative for fixed-step computational engineering where numerical stability and real-time execution are paramount.