Multilevel time-space adaptive finite volume method based on energy-norm error control for transient wave propagation

A multilevel time-space adaptive vertex-centered finite volume method is presented for transient wave propagation on unstructured meshes. The method employs a hybrid reconstruction using cell-center and edge-midpoint data to improve flux accuracy while preserving local conservation. Time integration is performed with a Runge-Kutta-Chebyshev scheme in which a single global number of stages is adjusted according to the stability requirement of the semi-discrete system, enabling efficient handling of moderately stiff systems. Adaptivity is driven by a posteriori error estimates in the energy norm and implemented through a multilevel refinement strategy that allows several mesh updates within one error-assessment cycle. Mesh conformity is maintained using newest-vertex bisection. The approach achieves substantial reductions in computational cost relative to single-level adaptive methods while maintaining high accuracy in capturing propagating wavefronts and material interfaces. Benchmark tests for radial and planar waves in complex geometries confirm that the method provides improved accuracy and efficiency compared with conventional explicit and implicit schemes of similar order.