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This paper presents a fully discrete numerical scheme for one-dimensional nonlocal wave equations and provides a rigorous theoretical analysis. To facilitate the spatial discretization, we introduce an auxiliary variable analogous to the gradient field in local discontinuous Galerkin (DG) methods for classical partial differential equations (PDEs) and reformulate the equation into a system of equations. The proposed scheme then uses a DG method for spatial discretization and the Crank-Nicolson method for time integration. We prove optimal L2 error convergence for both the solution and the auxiliary variable under a special class of radial kernels at the semi-discrete level. In addition, for general kernels, we demonstrate the asymptotic compatibility of the scheme, ensuring that it recovers the classical DG approximation of the local wave equation in the zero-horizon limit. Furthermore, we prove that the fully discrete scheme preserves the energy of the nonlocal wave equation. A series of numerical experiments are presented to validate the theoretical findings. 

Long-term fairness in sequential decision-making is critical yet challenging, as decisions at each time step influence future opportunities and outcomes, potentially exacerbating existing disparities over time. While existing methods primarily achieve fairness by directly adjusting decision models, in this work, we study a complementary perspective based on sequential algorithmic recourse, in which fairness is pursued through actionable interventions for individuals. We introduce Sequential Causal Algorithmic Recourse for Fairness (SCARF), a causally grounded framework that generates temporally coherent recourse trajectories by integrating structural causal modeling with sequential generative modeling. By explicitly incorporating both short-term and long-term fairness constraints, as well as practical budget limitations, SCARF generates personalized recourse plans that effectively mitigate disparities over multiple decision cycles. Through experiments on synthetic and semi-synthetic datasets, we empirically examine how different recourse strategies influence fairness dynamics over time, illustrating the trade-offs between short-term and long-term fairness under sequential interventions. The results demonstrate that SCARF provides a practical and informative framework for analyzing long-term fairness in dynamic decision-making settings.