Central-upwind schemes are Riemann-problem-solver-free Godunov-type finite-volume schemes, which are, in fact, non-oscillatory central schemes with a certain upwind flavor: derivation of the central-upwind numerical fluxes is based on the one-sided local speeds of propagation, which can be estimated using the largest and smallest eigenvalues of the Jacobian. I will introduce two new classes of central-upwind…

We consider weakly compressible flows coupled with a cloud system that models the dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a…

In this talk we present both numerical analysis and experiments to study a few basic computational issues in practice: (1) the numerical error one can achieve given a finite machine precision, (2) the learning dynamics and computation cost to achieve a given accuracy, and (3) stability with respect to perturbations. These issues are addressed for…

Physics-Informed Neural Networks (PINNs) have received much attention recently due to their potential for high-performance computations for complex physical systems, including data-based computing, systems with unknown parameters, and others. The idea of PINNs is to approximate the equations and boundary and initial conditions through a loss function for a neural network. PINNs combine the efficiency…