Table of Content

20 September 2024, Volume 6 Issue 3

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PREFACE

Preface

W. Boscheri, F. Chinesta, R. Loubere, S. Mishra, G. Puppo, M. Ricchiuto, C. -W. Shu

2024, 6(3):  1519-1520.  doi:10.1007/s42967-024-00434-6

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ORIGINAL PAPERS

Optimization of Artificial Viscosity in Production Codes Based on Gaussian Regression Surrogate Models

Vitaliy Gyrya, Evan Lieberman, Mark Kenamond, Mikhail Shashkov

2024, 6(3):  1521-1550.  doi:10.1007/s42967-023-00251-3

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To accurately model flows with shock waves using staggered-grid Lagrangian hydrodynamics, the artificial viscosity has to be introduced to convert kinetic energy into internal energy, thereby increasing the entropy across shocks. Determining the appropriate strength of the artificial viscosity is an art and strongly depends on the particular problem and experience of the researcher. The objective of this study is to pose the problem of finding the appropriate strength of the artificial viscosity as an optimization problem and solve this problem using machine learning (ML) tools, specifically using surrogate models based on Gaussian Process regression (GPR) and Bayesian analysis. We describe the optimization method and discuss various practical details of its implementation. The shock-containing problems for which we apply this method all have been implemented in the LANL code FLAG (Burton in Connectivity structures and differencing techniques for staggered-grid free-Lagrange hydrodynamics, Tech. Rep. UCRL-JC-110555, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1992, in Consistent finite-volume discretization of hydrodynamic conservation laws for unstructured grids, Tech. Rep. CRLJC-118788, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1994, Multidimensional discretization of conservation laws for unstructured polyhedral grids, Tech. Rep. UCRL-JC-118306, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1994, in FLAG, a multi-dimensional, multiple mesh, adaptive free-Lagrange, hydrodynamics code. In: NECDC, 1992). First, we apply ML to find optimal values to isolated shock problems of different strengths. Second, we apply ML to optimize the viscosity for a one-dimensional (1D) propagating detonation problem based on Zel’dovich-von NeumannDoring (ZND) (Fickett and Davis in Detonation: theory and experiment. Dover books on physics. Dover Publications, Mineola, 2000) detonation theory using a reactive burn model. We compare results for default (currently used values in FLAG) and optimized values of the artificial viscosity for these problems demonstrating the potential for significant improvement in the accuracy of computations.

Remapping Between Meshes with Isoparametric Cells: a Case Study

Mikhail Shashkov, Konstantin Lipnikov

2024, 6(3):  1551-1574.  doi:10.1007/s42967-023-00250-4

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We explore an intersection-based remap method between meshes consisting of isoparametric elements. We present algorithms for the case of serendipity isoparametric elements (QUAD8 elements) and piece-wise constant (cell-centered) discrete fields. We demonstrate convergence properties of this remap method with a few numerical experiments.

RKDG Methods with Multi-resolution WENO Limiters for Solving Steady-State Problems on Triangular Meshes

Jun Zhu, Chi-Wang Shu, Jianxian Qiu

2024, 6(3):  1575-1599.  doi:10.1007/s42967-023-00272-y

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In this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constIn this paper, we design high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (multi-resolution WENO) limiters to compute compressible steady-state problems on triangular meshes. A troubled cell indicator extended from structured meshes to unstructured meshes is constructed to identify triangular cells in which the application of the limiting procedures is required. In such troubled cells, the multi-resolution WENO limiting methods are used to the hierarchical L² projection polynomial sequence of the DG solution. Through using the RKDG methods with multi-resolution WENO limiters, the optimal high-order accuracy can be gradually reduced to first-order in the triangular troubled cells, so that the shock wave oscillations can be well suppressed. In steadystate simulations on triangular meshes, the numerical residual converges to near machine zero. The proposed spatial reconstruction methods enhance the robustness of classical DG methods on triangular meshes. The good results of these RKDG methods with multi-resolution WENO limiters are verified by a series of two-dimensional steady-state problems.

Second-Order Accurate Structure-Preserving Scheme for Solute Transport on Polygonal Meshes

Naren Vohra, Konstantin Lipnikov, Svetlana Tokareva

2024, 6(3):  1600-1628.  doi:10.1007/s42967-023-00289-3

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We analyze mimetic properties of a conservative finite-volume (FV) scheme on polygonal meshes used for modeling solute transport on a surface with variable elevation. Polygonal meshes not only provide enormous mesh generation flexibility, but also tend to improve stability properties of numerical schemes and reduce bias towards any particular mesh direction. The mathematical model is given by a system of weakly coupled shallow water and linear transport equations. The equations are discretized using different explicit cellcentered FV schemes for flow and transport subsystems with different time steps. The discrete shallow water scheme is well balanced and preserves the positivity of the water depth. We provide a rigorous estimate of a stable time step for the shallow water and transport scheme and prove a bounds-preserving property of the solute concentration. The scheme is second-order accurate over fully wet regions and first-order accurate over partially wet or dry regions. Theoretical results are verified with numerical experiments on rectangular, triangular, and polygonal meshes.

A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity

Lorenzo Micalizzi, Davide Torlo

2024, 6(3):  1629-1664.  doi:10.1007/s42967-023-00294-6

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The deferred correction (DeC) is an iterative procedure, characterized by increasing the accuracy at each iteration, which can be used to design numerical methods for systems of ODEs. The main advantage of such framework is the automatic way of getting arbitrarily high order methods, which can be put in the Runge-Kutta (RK) form. The drawback is the larger computational cost with respect to the most used RK methods. To reduce such cost, in an explicit setting, we propose an efficient modification: we introduce interpolation processes between the DeC iterations, decreasing the computational cost associated to the low order ones. We provide the Butcher tableaux of the new modified methods and we study their stability, showing that in some cases the computational advantage does not affect the stability. The flexibility of the novel modification allows nontrivial applications to PDEs and construction of adaptive methods. The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.

An Unconventional Divergence Preserving Finite-Volume Discretization of Lagrangian Ideal MHD

Walter Boscheri, Raphaël Loubère, Pierre-Henri Maire

2024, 6(3):  1665-1719.  doi:10.1007/s42967-023-00309-2

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We construct an unconventional divergence preserving discretization of updated Lagrangian ideal magnetohydrodynamics (MHD) over simplicial grids. The cell-centered finitevolume (FV) method employed to discretize the conservation laws of volume, momentum, and total energy is rigorously the same as the one developed to simulate hyperelasticity equations. By construction this moving mesh method ensures the compatibility between the mesh displacement and the approximation of the volume flux by means of the nodal velocity and the attached unit corner normal vector which is nothing but the partial derivative of the cell volume with respect to the node coordinate under consideration. This is precisely the definition of the compatibility with the Geometrical Conservation Law which is the cornerstone of any proper multi-dimensional moving mesh FV discretization. The momentum and the total energy fluxes are approximated utilizing the partition of cell faces into sub-faces and the concept of sub-face force which is the traction force attached to each sub-face impinging at a node. We observe that the time evolution of the magnetic field might be simply expressed in terms of the deformation gradient which characterizes the Lagrange-to-Euler mapping. In this framework, the divergence of the magnetic field is conserved with respect to time thanks to the Piola formula. Therefore, we solve the fully compatible updated Lagrangian discretization of the deformation gradient tensor for updating in a simple manner the cell-centered value of the magnetic field. Finally, the sub-face traction force is expressed in terms of the nodal velocity to ensure a semi-discrete entropy inequality within each cell. The conservation of momentum and total energy is recovered prescribing the balance of all the sub-face forces attached to the sub-faces impinging at a given node. This balance corresponds to a vectorial system satisfied by the nodal velocity. It always admits a unique solution which provides the nodal velocity. The robustness and the accuracy of this unconventional FV scheme have been demonstrated by employing various representative test cases. Finally, it is worth emphasizing that once you have an updated Lagrangian code for solving hyperelasticity you also get an almost free updated Lagrangian code for solving ideal MHD ensuring exactly the compatibility with the involution constraint for the magnetic field at the discrete level.

Failure-Informed Adaptive Sampling for PINNs, Part II: Combining with Re-sampling and Subset Simulation

Zhiwei Gao, Tao Tang, Liang Yan, Tao Zhou

2024, 6(3):  1720-1741.  doi:10.1007/s42967-023-00312-7

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This is the second part of our series works on failure-informed adaptive sampling for physic-informed neural networks (PINNs). In our previous work (SIAM J. Sci. Comput. 45: A1971–A1994), we have presented an adaptive sampling framework by using the failure probability as the posterior error indicator, where the truncated Gaussian model has been adopted for estimating the indicator. Here, we present two extensions of that work. The first extension consists in combining with a re-sampling technique, so that the new algorithm can maintain a constant training size. This is achieved through a cosine-annealing, which gradually transforms the sampling of collocation points from uniform to adaptive via the training progress. The second extension is to present the subset simulation (SS) algorithm as the posterior model (instead of the truncated Gaussian model) for estimating the error indicator, which can more effectively estimate the failure probability and generate new effective training points in the failure region. We investigate the performance of the new approach using several challenging problems, and numerical experiments demonstrate a significant improvement over the original algorithm.

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

Saray Busto, Michael Dumbser

2024, 6(3):  1742-1778.  doi:10.1007/s42967-023-00307-4

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In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

Projection-Based Dimensional Reduction of Adaptively Refined Nonlinear Models

Clayton Little, Charbel Farhat

2024, 6(3):  1779-1800.  doi:10.1007/s42967-023-00308-3

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Adaptive mesh refinement (AMR) is fairly practiced in the context of high-dimensional, mesh-based computational models. However, it is in its infancy in that of low-dimensional, generalized-coordinate-based computational models such as projection-based reducedorder models. This paper presents a complete framework for projection-based model order reduction (PMOR) of nonlinear problems in the presence of AMR that builds on elements from existing methods and augments them with critical new contributions. In particular, it proposes an analytical algorithm for computing a pseudo-meshless inner product between adapted solution snapshots for the purpose of clustering and PMOR. It exploits hyperreduction—specifically, the energy-conserving sampling and weighting hyperreduction method—to deliver for nonlinear and/or parametric problems the desired computational gains. Most importantly, the proposed framework for PMOR in the presence of AMR capitalizes on the concept of state-local reduced-order bases to make the most of the notion of a supermesh, while achieving computational tractability. Its features are illustrated with CFD applications grounded in AMR and its significance is demonstrated by the reported wallclock speedup factors.

Approximation Properties of Vectorial Exponential Functions

Christophe Buet, Bruno Despres, Guillaume Morel

2024, 6(3):  1801-1831.  doi:10.1007/s42967-023-00310-9

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This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference “Essentially Hyperbolic Problems: Unconventional Numerics, and Applications”. With respect to classical Finite Elements Methods, Trefftz methods are unconventional methods because of the way the basis functions are generated. Trefftz discontinuous Galerkin (TDG) methods have recently shown potential for numerical approximation of transport equations [6, 26] with vectorial exponential modes. This paper focuses on a proof of the approximation properties of these exponential solutions. We show that vectorial exponential functions can achieve high order convergence. The fundamental part of the proof consists in proving that a certain rectangular matrix has maximal rank.

Machine Learning Approaches for the Solution of the Riemann Problem in Fluid Dynamics: a Case Study

Vitaly Gyrya, Mikhail Shashkov, Alexei Skurikhin, Svetlana Tokareva

2024, 6(3):  1832-1859.  doi:10.1007/s42967-023-00334-1

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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 30%). 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.

Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral-Element Methods

Andrés M. Rueda-Ramírez, Benjamin Bolm, Dmitri Kuzmin, Gregor J. Gassner

2024, 6(3):  1860-1898.  doi:10.1007/s42967-023-00321-6

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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 highorder 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.

REVIEW ARTICLE

Physics-Based Active Learning for Design Space Exploration and Surrogate Construction for Multiparametric Optimization

Sergio Torregrosa, Victor Champaney, Amine Ammar, Vincent Herbert, Francisco Chinesta

2024, 6(3):  1899-1923.  doi:10.1007/s42967-023-00329-y

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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.

ORIGINAL PAPERS

Revisting High-Resolution Schemes with van Albada Slope Limiter

Jingcheng Lu, Eitan Tadmor

2024, 6(3):  1924-1953.  doi:10.1007/s42967-023-00348-9

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Slope limiters play an essential role in maintaining the non-oscillatory behavior of highresolution 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 onedimensional Total-Variation Diminishing (TVD) stability and demonstrate that it yields noticeable improvement in computation of one- and two-dimensional systems.

High Order ADER-IPDG Methods for the Unsteady Advection-Diffusion Equation

Michel Bergmann, Afaf Bouharguane, Angelo Iollo, Alexis Tardieu

2024, 6(3):  1954-1977.  doi:10.1007/s42967-023-00355-w

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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.

Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

Janina Bender, Philipp Öffner

2024, 6(3):  1978-2010.  doi:10.1007/s42967-024-00369-y

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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 twopoint 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.

New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with Uncertainties

Alina Chertock, Michael Herty, Arsen S. Iskhakov, Safa Janajra, Alexander Kurganov, Mária Lukáčová-Medvid'ová

2024, 6(3):  2011-2044.  doi:10.1007/s42967-024-00392-z

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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.