Mass-Lumped Virtual Element Method with Strong Stability-Preserving Runge-Kutta Time Stepping for Two-Dimensional Parabolic Problems

Abstract

This paper presents a mass-lumped Virtual Element Method (VEM) with explicit Strong Stability-Preserving Runge--Kutta (SSP-RK) time integration for two-dimensional parabolic problems on general polygonal meshes. A diagonal mass matrix is constructed via row-sum operations combined with flooring to ensure uniform positivity. Stabilization terms vanish identically under row summation, so the lumped weights derive solely from the $L^2$ projector and are computable through a small polynomial system at cost $\mathcal{O}(N_k^3)$ per element. The resulting lumped bilinear form satisfies $L^2$-equivalence with constants independent of the number of element edges, yielding a symmetric positive definite discrete inner product. A mesh-robust spectral estimate is established, showing that the largest eigenvalue of the discrete diffusion operator scales like $h^{-2}$, with constants depending only on the space dimension, polynomial degree, and mesh regularity. This yields the classical diffusion-type CFL condition $\Delta t=\mathcal{O}(h^2)$ for forward Euler stability and extends to higher-order SSP-RK schemes, ensuring the preservation of stability properties inherited from the forward Euler step. Numerical experiments on distorted quadrilateral, serendipity, and Voronoi meshes validate the theoretical predictions: for $k=1$, the lumped VEM attains optimal convergence rates, namely $\mathcal{O}(h)$ in the $H^1$-seminorm and $\mathcal{O}(h^2)$ in the $L^2$-norm, without degradation due to mesh distortion or diagonal mass approximation, while the SSP-RK methods remain stable under the predicted $\Delta t\propto h^2$ scaling. Additional tests on accuracy versus efficiency and on heterogeneous anisotropic diffusion further illustrate the practical competitiveness and robustness of the proposed formulation.