LDG Methods for an Ultrasonic Wave Equation With Weakly Singular Memory

ABSTRACT In this paper, we investigate local discontinuous Galerkin (LDG) schemes for solving an ultrasonic wave equation with weakly singular memory. We construct the LDG schemes using two distinct strategies. The first strategy follows the traditional construction approach, where only the spatial derivatives are coupled into a lower‐order system. The second strategy entails developing the LDG scheme by simultaneously addressing both temporal and spatial variables. The second strategy achieves superior alignment with time‐stepping schemes. To illustrate this, we utilize traditional time discretization schemes for the temporal discretization of the lower‐order system, resulting in four fully discrete schemes, accompanied by a comprehensive error analysis. The fully discrete LDG schemes are formulated by approximating the time‐fractional derivatives using the L1 approximation and Grünwald‐Letnikov formulas, while the first‐order time derivatives are discretized via Euler, BDF2, and Crank–Nicolson methods, respectively. Finally, extensive numerical experiments are conducted to validate the theoretical findings, demonstrating excellent agreement between computational results and analytical predictions, thereby confirming the method's effectiveness and high accuracy.