We present our results by using a machine learning (ML) approach for the solution of the Riemann problem for the Euler equations of fluid dynamics. The Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock tube. The solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics, such as finite-volume or discontinuous Galerkin methods. Therefore, a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial importance. The exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear equation. Prior to delving into the complexities of ML for the Riemann problem, we consider a much simpler formulation, yet very informative, problem of learning roots of quadratic equations based on their coefficients. We compare two approaches: (i) Gaussian process (GP) regressions, and (ii) neural network (NN) approximations. Among these approaches, NNs prove to be more robust and efficient, although GP can be appreciably more accurate (about ). We then use our experience with the quadratic equation to apply the GP and NN approaches to learn the exact solution of the Riemann problem from the initial data or coefficients of the gas equation of state (EOS). We compare GP and NN approximations in both regression and classification analysis and discuss the potential benefits and drawbacks of the ML approach.
We extend the monolithic convex limiting (MCL) methodology to nodal discontinuous Galerkin spectral-element methods (DGSEMS). The use of Legendre-Gauss-Lobatto (LGL) quadrature endows collocated DGSEM space discretizations of nonlinear hyperbolic problems with properties that greatly simplify the design of invariant domain-preserving high-resolution schemes. Compared to many other continuous and discontinuous Galerkin method variants, a particular advantage of the LGL spectral operator is the availability of a natural decomposition into a compatible subcell flux discretization. Representing a high-order spatial semi-discretization in terms of intermediate states, we perform flux limiting in a manner that keeps these states and the results of Runge-Kutta stages in convex invariant domains. In addition, local bounds may be imposed on scalar quantities of interest. In contrast to limiting approaches based on predictor-corrector algorithms, our MCL procedure for LGL-DGSEM yields nonlinear flux approximations that are independent of the time-step size and can be further modified to enforce entropy stability. To demonstrate the robustness of MCL/DGSEM schemes for the compressible Euler equations, we run simulations for challenging setups featuring strong shocks, steep density gradients, and vortex dominated flows.
The sampling of the training data is a bottleneck in the development of artificial intelligence (AI) models due to the processing of huge amounts of data or to the difficulty of access to the data in industrial practices. Active learning (AL) approaches are useful in such a context since they maximize the performance of the trained model while minimizing the number of training samples. Such smart sampling methodologies iteratively sample the points that should be labeled and added to the training set based on their informativeness and pertinence. To judge the relevance of a data instance, query rules are defined. In this paper, we propose an AL methodology based on a physics-based query rule. Given some industrial objectives from the physical process where the AI model is implied in, the physics-based AL approach iteratively converges to the data instances fulfilling those objectives while sampling training points. Therefore, the trained surrogate model is accurate where the potentially interesting data instances from the industrial point of view are, while coarse everywhere else where the data instances are of no interest in the industrial context studied.
Slope limiters play an essential role in maintaining the non-oscillatory behavior of high-resolution methods for nonlinear conservation laws. The family of minmod limiters serves as the prototype example. Here, we revisit the question of non-oscillatory behavior of high-resolution central schemes in terms of the slope limiter proposed by van Albada et al. (Astron Astrophys 108: 76–84, 1982). The van Albada (vA) limiter is smoother near extrema, and consequently, in many cases, it outperforms the results obtained using the standard minmod limiter. In particular, we prove that the vA limiter ensures the one-dimensional Total-Variation Diminishing (TVD) stability and demonstrate that it yields noticeable improvement in computation of one- and two-dimensional systems.
We present a high-order Galerkin method in both space and time for the 1D unsteady linear advection-diffusion equation. Three Interior Penalty Discontinuous Galerkin (IPDG) schemes are detailed for the space discretization, while the time integration is performed at the same order of accuracy thanks to an Arbitrary high order DERivatives (ADER) method. The orders of convergence of the three ADER-IPDG methods are carefully examined through numerical illustrations, showing that the approach is consistent, accurate, and efficient. The numerical results indicate that the symmetric version of IPDG is typically more accurate and more efficient compared to the other approaches.
In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos (gPC) approach which we consider in the following manuscript. We apply the stochastic Galerkin (SG) method to the stochastic SW equations. Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore. The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations. By transforming the system using Roe variables, the hyperbolicity can be ensured and an entropy-entropy flux pair is known from a recent investigation by Gerster and Herty (Commun. Comput. Phys. 27(3): 639–671, 2020). We use this pair and determine a corresponding entropy flux potential. Then, we construct entropy conservative numerical two-point fluxes for this augmented system. By applying these new numerical fluxes in a nodal DG spectral element method (DGSEM) with flux differencing ansatz, we obtain a provable entropy conservative (dissipative) scheme. In numerical experiments, we validate our theoretical findings.
In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory (WENO) interpolations in (multidimensional) random space combined with second-order piecewise linear reconstruction in physical space. Compared with spectral approximations in the random space, the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy. The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations. In the latter case, the methods are also proven to be well-balanced and positivity-preserving.