Numerical analysis of spatiotemporal high-index saddle dynamics for finding multiple solutions of semilinear elliptic problems

Abstract: This paper presents a rigorous numerical framework for computing multiple solutions of semilinear elliptic problems by spatiotemporal high-index saddle dynamics (HiSD), which extends the traditional HiSD to the continuous-in-space setting, explicitly incorporating spatial differential operators. To enforce the Stiefel manifold constraint without introducing the analytical complications of retraction-based updates, we design a fully discrete retraction-free orthonormality-preserving scheme for spatiotemporal HiSD. This scheme exhibits favorable structural properties that substantially reduce the difficulties arising from coupling and gradient nonlinearities in spatiotemporal HiSD. Exploiting these properties, we establish gradient stability and error estimates, which consequently ensure the preservation of the Morse index for the computed saddle points. The framework is further extended to the semilinear advection-reaction-diffusion equation. Numerical experiments demonstrate the efficiency of the proposed method in finding multiple solutions and constructing the solution landscape of semilinear elliptic problems. To the best of our knowledge, this work presents the first rigorous full space--time accuracy analysis of the HiSD system. It reveals intrinsic connections between saddle-search algorithms and numerical methods for PDEs, enhancing their mutual compatibility for a broad range of problems.