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Description

Improving the accuracy of numerical modeling for compressible fluid flows remains a critical area of research. This two-part presentation explores novel solutions to challenges in simulating compressible flows in relativistic and Newtonian regimes. In Part 1, we review a persistent numerical hurdle: resolving relativistic shocks in a physically consistent manner. Even in 1D simulations, existing High-Resolution Shock-Capturing (HRSC) approaches often fail to achieve grid convergence without impractically fine resolutions—particularly when relativistic tangential velocities are present. To address these deficiencies, we introduce a new hybrid method that significantly improves the fidelity of relativistic shock-tube solutions, delivering sharply resolved shock and contact discontinuities. In Part 2, we introduce a novel, automatic shock-capturing scheme based on Gaussian Process (GP) modeling for conventional non-relativistic Newtonian flows. This method leverages GP regression to compute optimized, high-order solutions in smooth regions for maximum accuracy. Simultaneously, it ensures non-oscillatory stability near discontinuities. We design this auto-optimizing GP scheme using a unique optimization of GP kernel hyperparameters and stencil-size variations. This is achieved by maximizing the marginal log-likelihood, allowing the scheme to adaptively balance precision and stability.