Shock-capturing Research Articles

Comparing Discontinuous Galerkin Shock-Capturing Techniques Applied to Inviscid Three-Dimensional Hypersonic Flows

This paper considers the performance of various shock-capturing schemes when simulating three-dimensional, hypersonic flows using the nodal discontinuous Galerkin (DG) finite element method on unstructured, hexahedral meshes. Simulations use a new code, Cartablanca++, which is verified using the method of manufactured solutions. Three shock-capturing techniques are compared: artificial viscosity (AV), slope limiting, and subcell finite volume limiting. Three test cases are considered, including a shock tube (one-dimensional), a reflecting shock (two-dimensional), and an inclined cylinder with a hemispherical endcap (three-dimensional). The AV formulation was not robust in the sense that it could not maintain pressure positivity after initialization from freestream conditions in the final three-dimensional test case. The slope and subcell limiters performed well in all simulations. Both techniques robustly captured strong shock waves while still benefiting from the use of high-order polynomials. The targeted application of the slope limiter prevented residual convergence to machine precision, while the subcell limiter could achieve residual convergence. The mixed DG/finite volume formulation, inherent to the subcell limiting scheme, appears sensitive to the inviscid flux function. Future work will consider modifications to reduce this sensitivity. Additionally, modifications to the shock detection techniques would improve performance for both the slope and subcell limiters.

OpenSBLI v3.0: High-fidelity multi-block transonic aerofoil CFD simulations using domain specific languages on GPUs

OpenSBLI is an automatic code-generation framework for compressible Computational Fluid Dynamics (CFD) simulations on heterogeneous computing architectures (previous release: Lusher et al. (2021) [4]). OpenSBLI is coupled to the Oxford Parallel Structured (OPS) Domain Specific Language (DSL), which uses source-to-source translation to enable parallel execution of the code on large-scale supercomputers, including multi-GPU clusters. To date, OpenSBLI has largely been applied to compressible turbulence and shock-wave/boundary-layer interactions on very simple geometries comprised of single mesh blocks with essentially orthogonal grid lines. OpenSBLI has been extended in this new release to target strongly curvilinear cases, including transonic aerofoils using multi-block grids. In addition to multi-block mesh support, more efficient numerical shock-capturing methods and filters have been added to the codebase. Improvements to post-processing, reduced-dimension data output, and coupling to a modal decomposition library are also included. A set of validation cases are presented to showcase the new code features. Furthermore, state-of-the-art wide-span transonic aerofoil simulations on up to N=2.5×109 grid points demonstrate that wider aspect ratios can alter buffet predictions and increase the regularity of the low-frequency shock oscillations by accommodating fully-developed trailing edge flow separation. Spectral Proper Orthogonal Decomposition (SPOD) analysis showed that overly-narrow aerofoil simulations contain additional domain-dependent energy content at a Strouhal number of St≈3 associated with wake modes.

Novel spectral methods for shock capturing and the removal of tygers in computational fluid dynamics

Spectral methods yield numerical solutions of the Galerkin-truncated versions of nonlinear partial differential equations (PDEs) involved especially in fluid dynamics. In the presence of discontinuities, such as shocks, spectral approximations develop Gibbs oscillations near the discontinuity. This causes the numerical solution to deviate quickly from the true solution. For spectral approximations of the inviscid Burgers equation in one-dimension (1D), nonlinear wave resonances lead to the formation of localised oscillatory structures called tygers in well-resolved areas of the flow, far from the shock. Recently, the author of [1] has proposed novel spectral relaxation (SR) and spectral purging (SP) schemes for the removal of tygers and Gibbs oscillations in spectral approximations of nonlinear conservation laws. For the 1D inviscid Burgers equation, it is shown in [1] that the novel SR and SP approximations of the solution converge strongly in L2 norm to the entropic weak solution, under an appropriate choice of kernels and related parameters. In this work, we carry out a detailed numerical investigation of SR and SP schemes when applied to the 1D inviscid Burgers equation and report the efficiency of shock capture and the removal of tygers. We then extend our study to systems of nonlinear hyperbolic conservation laws, such as the 2×2 system of the shallow water equations and the standard 3×3 system of 1D compressible Euler equations. For the latter, we generalise the implementation of SR methods to non-periodic problems using Chebyshev polynomials. We then turn to singular flow in the 1D wall approximation of the 3D-axisymmetric wall-bounded incompressible Euler equation [2]. Here, in order to determine the blowup time of the solution, we compare the decay of the width of the analyticity strip, obtained from the pure pseudospectral method (PPS), with the improved estimate obtained using the novel spectral relaxation scheme.

A novel finite-difference converged ENO scheme for steady-state simulations of Euler equations

The high-order shock-capturing scheme is critical for compressible fluid simulations, in particular for cases where strong shockwaves are present. However, for the steady-state solution of Euler equations, the high-order shock-capturing schemes usually suffer from the difficulty that the numerical residual hangs at a relatively high level instead of converging to machine zero. In this paper, a new paradigm of finite-difference converged ENO (CENO) scheme for steady-state simulations of Euler equations is proposed. Different from the existing methods dedicating to eliminating the slight post-shock oscillation, the unsteady effect of shock-induced numerical instability on the nonlinear weighting process as well as the resultant variation of the numerical formula for the spatially fixed cell interface flux at different time instants are demonstrated to be the core contributors to the non-convergence of high-order shock-capturing schemes. Unlike classical ENO/WENO/TENO schemes, which generate the final reconstruction based only on spatial information, evolutionary information is also exploited by a novel converged stencil selection strategy in the present scheme. With this strategy, the unsteady effect of shock-induced numerical instability on the nonlinear weighting process is incorporated and eliminated by a tailored adaptive scale-separation algorithm, while shock-capturing capabilities are not sacrificed. A set of well-established and newly designed challenging benchmark simulations involving strong shockwaves, contact discontinuities and rarefaction waves demonstrates that the new fifth-order CENO5 scheme features superior convergence properties and achieves the essentially non-oscillation property better when compared to the robust TENO5 scheme and the state-of-the-art WENO-type schemes designed for steady-state problems.

Unconditionally optimal high-order weighted compact nonlinear schemes with sharing function for Euler equations

Weighted compact nonlinear schemes (WCNS) are a type of high-order shock-capturing finite difference schemes commonly used in various applications. Their spatial discretizations involve a nonlinear interpolation step and a linear difference step. However, the nonlinear interpolation step requires significantly more computational resources compared to the linear difference step. Therefore, simplifying the interpolation step is an effective way to improve the efficiency of these schemes. In this paper, we propose a new approach to construct WCNS schemes based on the sharing function for Euler equations. This approach uses a set of common nonlinear interpolation weights for different components, resulting in a significant improvement in efficiency compared to the original WCNS schemes that use different sets of weights for each component. To ensure accuracy at critical points of any orders, we employ nonlinear weights that guarantee unconditionally optimal high order. Additionally, this new approach may reduce oscillations caused by non-characteristic interpolation. Furthermore, we also develop a new shock detector using the sharing function, enabling us to employ characteristic interpolation for nonsmooth regions and linear component-wise interpolation for the rest. We validate the proposed schemes based on the sharing function through numerical examples of one- and two-dimensional Euler equations, demonstrating their effectiveness in terms of shock-capturing capability and efficiency.

P adé: A Code for Protoplanetary Disk Turbulence Based on Padé Differencing

The Padé code has been developed to treat hydrodynamic turbulence in protoplanetary disks. It solves the compressible equations of motion in cylindrical coordinates. Derivatives are computed using nondiffusive and conservative fourth-order Padé differencing, which has higher resolving power compared to both dissipative shock-capturing schemes used in most astrophysics codes, as well as nondiffusive central finite-difference schemes of the same order. The fourth-order Runge–Kutta method is used for time stepping. A previously reported error-corrected Fargo approach is used to reduce the time step constraint imposed by rapid Keplerian advection. Artificial bulk viscosity is used when shock capturing is required. Tests for correctness and scaling with respect to the number of processors are presented. Finally, efforts to improve efficiency and accuracy are suggested.

Steady-state simulation of Euler equations by the discontinuous Galerkin method with the hybrid limiter

In the realm of steady-state solutions of Euler equations, the challenge of achieving convergence of residue close to machine zero has long plagued high-order shock-capturing schemes, particularly in the presence of strong shock waves. In response to this issue, this paper presents a hybrid limiter designed specifically for the discontinuous Galerkin (DG) method which incorporates a pseudo-time marching method. This hybrid limiter, initially introduced in DG methods for unsteady problems, preserves the local data structure while enhancing resolution through effective and precise shock capturing. Notably, for the steady problem, the hybridization of the DG solution with the cell average is employed, deviating from the low-order limited DG solution typically used for unsteady problems. This approach seamlessly integrates the previously distinct troubled cell indicator and limiter components resulting in a more cohesive and efficient limiter. It obviates the need for characteristic decomposition and intercell communication, leading to substantial reductions in computational costs and enhancement in parallel efficiency. Numerical experiments are presented to demonstrate the robust performance of the hybrid limiter for steady-state Euler equations both on the structured and unstructured meshes.

URANOS-2.0: Improved performance, enhanced portability, and model extension towards exascale computing of high-speed engineering flows

We present URANOS-2.0, the second major release of our massively parallel, GPU-accelerated solver for compressible wall flow applications. This latest version represents a significant leap forward in our initial tool, which was launched in 2023 (De Vanna et al. [1]), and has been specifically optimized to take full advantage of the opportunities offered by the cutting-edge pre-exascale architectures available within the EuroHPC JU. In particular, URANOS-2.0 emphasizes portability and compatibility improvements with the two top-ranked supercomputing architectures in Europe: LUMI and Leonardo. These systems utilize different GPU architectures, AMD and NVIDIA, respectively, which necessitates extensive efforts to ensure seamless usability across their distinct structures. In pursuit of this objective, the current release adheres to the OpenACC standard. This choice not only facilitates efficient utilization of the full potential inherent in these extensive GPU-based architectures but also upholds the principles of vendor neutrality, a distinctive characteristic of URANOS solvers in the CFD solvers' panorama. However, the URANOS-2.0 version goes beyond the goals of improving usability and portability; it introduces performance enhancements and restructures the most demanding computational kernels. This translates into a 2× speedup over the same architecture. In addition to its enhanced single-GPU performance, the present solver release demonstrates very good scalability in multi-GPU environments. URANOS-2.0, in fact, achieves strong scaling efficiencies of over 80% across 64 compute nodes (256 GPUs) for both LUMI and Leonardo. Furthermore, its weak scaling efficiencies reach approximately 95% and 90% on LUMI and Leonardo, respectively, when up to 256 nodes (1024 GPUs) are considered. These significant performance advancements position URANOS-2.0 as a state-of-the-art supercomputing platform tailored for compressible wall turbulence applications, establishing the solver as an integrated tool for various aerospace and energy engineering applications, which can span from direct numerical simulations, wall-resolved large eddy simulations, up to most recent wall-modeled large eddy simulations. Program summaryProgram title: Unsteady Robust All-around Navier-StOkes Solver (URANOS)CPC Library link to program files:https://doi.org/10.17632/pw5hshn9k6.2Developer's repository link:https://github.com/uranos-gpu/uranos-gpu, https://github.com/uranos-gpu/uranos-gpu/tree/v2.0Licensing provisions: BSD License 2.0Programming language: Modern Fortran, OpenACC, MPINature of problem: Solving the compressible Navier-Stokes equations in a three-dimensional Cartesian framework.Solution method: Convective terms are treated with high-resolution shock-capturing schemes. The system dynamics is advanced in time with a three-stage Runge-Kutta method. Parallelization adopts MPI+OpenACC.

Convergence towards High-Speed Steady States Using High-Order Accurate Shock-Capturing Schemes

Creating time-marching unsteady governing equations for a steady state in high-speed flows is not a trivial task. Residue convergence in time cannot be achieved when using most low- and high-order spatial discretization schemes. Recently, high-order, weighted, essentially non-oscillatory schemes have been specially designed for steady-state simulations. They have been shown to be capable of achieving machine precision residues when simulating the Euler equations under canonical coordinates. In the present work, we review these schemes and show that they can also achieve machine residues when simulating the Navier–Stokes equations under generalized coordinates. This is carried out by considering three supersonic flows of perfect fluids, namely the flow upstream a cylinder, the flow over a blunt wedge, and the flow over a compression ramp.

Numerical stability analysis of Godunov-type schemes for high Mach number flow simulations

Modern shock-capturing schemes often suffer from numerical shock instabilities when simulating strong shocks, limiting their application in supersonic or hypersonic flow simulations. In the current study, we devote our efforts to investigating the shock instability problem for second-order schemes, which has not gotten enough attention in previous research but is crucial to address. To this end, we develop the matrix stability analysis method for the finite-volume Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) approach, taking into account the influence of reconstruction. With the help of this newly developed method, the shock instability of second-order schemes is investigated quantitatively and efficiently. The results demonstrate that when second-order schemes are employed, whether shock instabilities will occur is closely related to the property of Riemann solvers, just like the first-order case. However, enhancing spatial accuracy still impacts the shock instability problem, and the impact can be categorized into two types depending on the dissipation of Riemann solvers. Furthermore, the research emphasizes the impact of the numerical shock structure, highlighting both its role as the source of instability and the influence of its state on the occurrence of instability. One of the most significant contributions of this study is the confirmation of the multidimensional coupled nature of shock instability. Both one-dimensional and multidimensional instabilities are proven to influence the instability problem, and they have different properties. Moreover, this paper reveals that increasing the aspect ratio and distortion angle of the computational grid can help mitigate shock instabilities. The current work provides an effective tool for quantitatively investigating the shock instabilities for second-order schemes, revealing the inherent mechanism and thereby contributing to the elimination of shock instability.

A new high-order shock-capturing TENO scheme combined with skew-symmetric-splitting method for compressible gas dynamics and turbulence simulation

The high-order shock-capturing scheme is one of the main building blocks for the simulation of the compressible fluid characterized by strong shockwaves and broadband length scales. However, the classical shock-capturing scheme fails to perform long-time stable and non-dissipative simulations since the quadratic invariants associated with the conservation equations cannot be conserved as a result of the inherent numerical dissipation. Additionally, the overall computational cost for classical shock-capturing schemes is quite expensive as a result of the time-consuming local characteristic decomposition and the nonlinear-weights computing process. In this work, based on a new efficient discontinuity indicator, which distinguishes the non-smooth high-wavenumber fluctuations and discontinuities from smooth scales in the wavenumber space, a paradigm of high-order shock-capturing scheme by recasting the non-dissipative skew-symmetric-splitting method with newly optimized dispersion property for smooth flow scales and invoking the nonlinear targeted ENO (TENO) schemes for non-smooth discontinuities is proposed. The resulting TENO-S scheme not only successfully performs long-time stable computations for smooth flows without numerical dissipation, but also recovers the robust shock-capturing capabilities with adaptive numerical dissipation. Without the necessity of parameter tuning case by case, extensive benchmark simulations involving a wide range of flow length scales and strong discontinuities demonstrate that the proposed TENO-S scheme performs significantly better than the straightforward deployment of WENO/TENO-family schemes with better spectral property and higher computational efficiency.

A general multi-objective Bayesian optimization framework for the design of hybrid schemes towards adaptive complex flow simulations

Achieving accuracy with underresolved simulation of complex compressible flows with multiscale flow structures is a challenge. Either the numerical dissipation or the resolution and thereby the numerical cost is impractically high. Also, in the design of numerical solvers, the application of a solver for specific flow classes is balanced by robustness allowing the study of a broad range of flows. In this study, we propose a hybrid fifth-order targeted essentially non-oscillatory (TENO5)-based scheme tailored to optimally simulate compressible flows with underresolved dilatational and vortical multiscale structures. For optimal design, three data-driven objectives are defined. A novel objective that derives from the numerical dissipation rate analysis is a key element to deal with underresolved complex flows in practical applications. The optimization process employs a multi-objective Bayesian optimization framework with an expected hypervolume improvement and three flow configurations representative for a broad range of two- and three-dimensional flows with genuine and non-genuine subgrid scales. The optimized hybrid scheme is validated by comparing with shock-capturing schemes of the weighted essentially non-oscillatory (WENO)- and TENO- families with flows of complex shock interactions, Kelvin-Helmholtz instabilities, shock-vortex interactions, vortical and turbulent flows

A dynamic one-dimensional model for simulating unsteady air-water stratified flow in sewer pipes.

Ventilation is paramount in sanitary and stormwater sewer systems to mitigate odor problems and avert pressure surges. Existing numerical models have constraints in practical applications in actual sewer systems due to insufficient airflow modeling or suitability only for steady-state conditions. This research endeavors to formulate a mathematical model capable of accurately simulating various operational conditions of sewer systems under the natural ventilation condition. The dynamic water flow is modeled using a shock-capturing MacCormack scheme. The dynamic airflow model amalgamates energy and momentum equations, circumventing laborious pressure iteration computations. This model utilizes friction coefficients at interfaces to enhance the description of the momentum exchange in the airflow and provide a logical explanation for air pressure. A systematic analysis indicates that this model can be easily adapted to include complex boundary conditions, facilitating its use for modeling airflow in real sewer networks. Furthermore, this research uncovers a direct correlation between the air-to-water flow rate ratio and the filling ratio under natural ventilation conditions, and an empirical formula encapsulating this relationship is derived. This finding offers insights for practical engineering applications.

Shock-capturing PID controller for high-order methods with data-driven gain optimization

We present a novel shock-capturing strategy for high-order methods including discontinuous Galerkin (DG) method with data-driven gain optimization. Inspired by the classical control theory, we utilize a proportional–integral–derivative (PID) controller for capturing shock waves with monotonic subcell distributions. The proposed closed-loop control system for shock-capturing, named shock-capturing PID controller (SPID), consists of two key elements: error estimation based on the multi-dimensional limiting strategies (hMLP and hMLP_BD) and shock stabilization using the Laplacian artificial viscosity (LAV). The two elements are combined in a complementary manner to maximize the advantages of limiting strategy and artificial viscosity while overcoming each weakness. First, based on the multi-dimensional limiting process (MLP) condition and the troubled-boundary detector, the estimated error gives a signal to the SPID how much flow variables stray out of monotonic shock profiles. Second, the SPID estimates the amount of artificial viscosity to stabilize the target shock wave and superimposes numerical diffusion to the governing equations in the form of LAV. Each control action of the SPID (i.e., proportional, integral, and derivative) has a distinct role in capturing and stabilizing shock waves. The proportional action determines a minimal amount of background artificial viscosity. The integral action reinforces a shock-stabilizing numerical diffusion where the artificial viscosity by the proportional action alone is insufficient. The derivative action damps out sudden rises of error and spurious oscillations. The SPID incorporates the gain parameters that regulate the impact of each control action, and each gain parameter is determined via a surrogate-based optimization approach utilizing artificial neural networks (ANN). The SPID with the data-driven gain parameters is verified and validated by conducting extensive numerical tests and by comparing the results to other shock-capturing methods (hMLP, hMLP_BD, and Laplacian artificial viscosity). The numerical results demonstrate the excellent performance of SPID in terms of capturing shock waves and stabilizing shock-induced oscillations. Moreover, the SPID successfully preserves unsteady turbulent eddies for large-eddy simulations (LES) of subsonic and supersonic flows and improves convergence characteristics of steady transonic/supersonic flows.

Improvement of Z-Weighted Function Based on Fifth-Order Nonlinear Multi-Order Weighted Method for Shock Capturing of Hyperbolic Conservation Laws.

Based on a 5-point stencil and three 3-point stencils, a nonlinear multi-order weighted method adaptive to 5-3-3-3 stencils for shock capturing is presented in this paper. The form of the weighting function is the same as JS (Jiang-Shu) weighting; however, the smoothness indicator of the 5-point stencil adopts a special design with a higher-order leading term similar to the τ in Z weighting. The design maintains that the nonlinear weights satisfy sufficient conditions for the scheme to avoid degradation even near extreme points. By adjusting the linear weights to a specific value and using the τ in Z weighting, the method can be degraded to Z weighting. Analysis of linear weights shows that they do not affect the accuracy in the smooth region, and they can also adjust the resolution and discontinuity-capturing capability. Numerical tests of different hyperbolic conservation laws are conducted to test the performance of the newly designed nonlinear weights based on the weighted compact nonlinear scheme. The numerical results show that there are no obvious oscillations near the discontinuity, and the resolution of both the discontinuity and smooth regions is better than that of Z weights.

On the accuracy of shock-capturing schemes when calculating Cauchy problems with periodic discontinuous initial data

Abstract We study the accuracy of shock-capturing schemes for the shallow water Cauchy problems with piecewise smooth discontinuous initial data. We consider the second order balance-characteristic (CABARETM) scheme, the third order finite-difference Rusanov–Burstein–Mirin (RBM) scheme and the fifth order in space, the third order in time weighted essentially non-oscillatory (WENO5) scheme. We have shown that the maximum loss of accuracy occurs in the centered rarefaction waves of the exact solutions, where all these schemes have the first order of convergence and fairly close values of the numerical disbalances (errors), regardless of their formal approximation order on the smooth solutions. In the same time, inside the shock influence areas the considered schemes can have different convergence orders and, as a result, significantly different accuracy. In particular, when solving the Cauchy problem with periodic initial data, when the exact solution has no centered rarefaction waves, the RBM scheme has a significantly higher accuracy inside the shock influence areas, compared to the CABARETM and WENO5 schemes. It means that the combined scheme, in which the RBM scheme is a basic scheme and the CABARETM scheme is an internal one, can be effectively used to compute weak solutions of such type Cauchy problems.

A high-resolution meshfree particle method for numerical investigation of second-order macroscopic pedestrian flow models

Recent advancements in empirical observations of human psychological responses have revealed that the governing rules of human behavioral aspects can not only be predicted with adequate deterministic precision but also forged into mathematical formulations. These modeling studies enable crisis managers with reliable simulation tools to gain insights into critical facets of crowd disasters and enhance their decision-making abilities to facilitate safe egress in emergency situations. The macroscopic scale of representation in this context is often invoked to comprehend large-scale collective pattern formation in vast built environments, owing to its computational viability. The associated governing equations are typically constructed in a system of hyperbolic conservation law form, and conducting simulations in real-life complex geometric configurations can pose computational challenges. This article puts forward a high-resolution, shock-capturing meshfree particle method in an Eulerian framework for the numerical approximation of several widely adopted macroscopic pedestrian flow models. It serves as a superior alternative to traditional mesh-based methods, eliminating the need for specific mesh structures and connectivity between them. A classical Generalized Finite Difference Method (GFDM) in conjunction with several consistency conditions on spatial derivative coefficients is adopted to obtain a non-oscillatory and fairly conservative Godunov-type discretization of the governing system. The intercell flux is evaluated using a suitable Riemann solver, and a slope-limited reconstruction procedure of the corresponding Riemann states, adapted to arbitrary point distribution, is employed for improved accuracy. Moreover, the preferred direction of human motion is determined from a local density-dependent eikonal-type equation, which is solved using a generalized version of the Fast Marching Method. A thorough numerical investigation of three well-known second-order macroscopic models, namely Payne-Whitham (PW), Aw-Rascle (AR), and Aw-Rascle-Zhang (ARZ) is carried out through two hypothetical scenarios of crowd evacuation, not only to demonstrate the accuracy and applicability of the proposed approach but also to deepen understanding of their distinctive characteristics. A comparison of the flow flux with the fundamental diagram suggests that the density profile obtained with the ARZ model is equivalent to Hughes' first-order model. In contrast, the PW and AR models have proven to effectively replicate complex, unstable pedestrian phenomena such as stop-and-go waves. Finally, Helbing's counter-intuitive idea that strategically positioned barriers upstream of a bottleneck can essentially expedite evacuation is numerically investigated across various parameter choices.

Quasi all-speed schemes for magnetohydrodynamic flows in a wide range of Mach numbers

We present novel numerical schemes for ideal magnetohydrodynamic (MHD) simulations aimed at enhancing stability against numerical shock instability and improving the accuracy of low-speed flows in multidimensions. Stringent benchmark tests confirm that our scheme is more robust against numerical shock instability and is more accurate for low-speed, nearly incompressible flows than conventional shock-capturing schemes. Our scheme is a promising tool for tackling MHD systems, including both high and low Mach number flows.

Version [1.1]—[MSAT: Matrix stability analysis tool for shock-capturing schemes]

The numerical shock instability problem significantly limits the application of the finite volume method in hypersonic flow simulations, especially for high-order schemes, which will encounter more severe shock instability problems. In the previous work, the MSAT (matrix stability analysis tool) was introduced to quantitatively analyze the numerical shock instability problem. However, the utilization of the three-point reconstruction stencil in the original version of MSAT presents challenges in conducting investigations into high-order schemes. The current work provides an extension of MSAT by extending the reconstruction stencil to five points, a challenging but necessary endeavor for high-order schemes. This extension enables MSAT to provide analyses for schemes with high-order reconstruction methods. Furthermore, the updated MSAT can be utilized to investigate the impact of characteristic variable reconstruction on the shock instability problem. Results demonstrate the reliability of the updated MSAT and reveal novel characteristics in the shock instability of high-order schemes. Consequently, the updated MSAT will contribute to the investigation of shock instabilities for high-order schemes, ultimately aiding in mitigating this challenging problem.

Gradient-annihilated PINNs for solving Riemann problems: Application to relativistic hydrodynamics

We present a novel methodology based on Physics-Informed Neural Networks (PINNs) for solving systems of partial differential equations admitting discontinuous solutions. Our method, called Gradient-Annihilated PINNs (GA-PINNs), introduces a modified loss function that forces the model to partially ignore high-gradients in the physical variables, achieved by introducing a suitable weighting function. The method relies on a set of hyperparameters that control how gradients are treated in the physical loss. The performance of our methodology is demonstrated by solving Riemann problems in special relativistic hydrodynamics, extending earlier studies with PINNs in the context of the classical Euler equations. The solutions obtained with the GA-PINN model correctly describe the propagation speeds of discontinuities and sharply capture the associated jumps. We use the relative l2 error to compare our results with the exact solution of special relativistic Riemann problems, used as the reference “ground truth”, and with the corresponding error obtained with a second-order, central, shock-capturing scheme. In all problems investigated, the accuracy reached by the GA-PINN model is comparable to that obtained with a shock-capturing scheme, achieving a performance superior to that of the baseline PINN algorithm in general. An additional benefit worth stressing is that our PINN-based approach sidesteps the costly recovery of the primitive variables from the state vector of conserved variables, a well-known drawback of grid-based solutions of the relativistic hydrodynamics equations. Due to its inherent generality and its ability to handle steep gradients, the GA-PINN methodology discussed in this paper could be a valuable tool to model relativistic flows in astrophysics and particle physics, characterized by the prevalence of discontinuous solutions.