General Purpose Alternative Finite Difference WENO (AFD-WENO) for Conservative Systems and Systems with Non-Conservative Products

By Dinshaw S. Balsara (Physics and ACMS, Univ. of Notre Dame)

2 PM, 9 January 2024, Auditorium, TIFR-CAM

In their landmark sequence of papers (Shu and Osher 1988, 1989) the authors presented two highly efficient finite difference WENO schemes. The latter finite difference WENO scheme (Shu and Osher 1989) has become wildly successful and garnered thousands of citations. We call that the FD-WENO scheme. However, in Shu and Osher (1988) they also presented an alternative finite difference WENO (AFD-WENO) scheme which was slower to catch on. We explain why that scheme was slower to catch on - it is because all ingredients that are needed to make a production code out of AFD-WENO were not available at that time. Besides, the scheme was not easy to understand at the time of its initial presentation. We demystify the AFD-WENO algorithm in this talk.

In this talk we explain why the AFD-WENO scheme, nevertheless, had several significant advantages, if it could be developed into an automated algorithm for production codes. This talk is devoted to developing AFD-WENO into a simple algorithm that is easily explained to others and also easily implemented in production codes. To reach that goal, we had to make several algorithmic innovations which we explain here.

The original FD-WENO schemes were also only viable for conservation laws. But the field has moved on and it is very normal for scientists and engineers to discover hyperbolic PDE systems that have non-conservative products, often with stiff source terms. To accommodate such PDE systems, we present the first of its kind AFD-WENO scheme that can retain strict conservation when the PDE is conservative, and yet, accommodate non-conservative products. This vastly expands the class of PDEs that can be treated with AFD-WENO schemes. Several examples are demonstrated in this talk.

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Metric-field based mesh adaptation for compressible flow simulations

Aravind Balan

Dept. of Aerospace Engg.

Indian Institute of Science

Bangalore14 November 2023 at 4:00 PM

TIFR-CAM, Bangalore and on ZoomAbstract: For resolving anisotropic flow features such as shocks, boundary layers, etc., in compressible flow simulations, anisotropic (stretched) meshes are much more efficient than isotropic meshes. Metric-field based mesh generation provides a suitable framework to incorporate anisotropic features of the solutions on the meshes. In this framework, the meshes are described by metric tensors that encode size, anisotropy and the orientation of the simplex mesh elements. For discontinuous Galerkin methods, discretization error control can be done using either h-adaptation, or hp-adaptation. Efficient h- and hp-adaptation methods based on metric fields are developed for discontinuous Galerkin methods for solving compressible flow simulations. The effectiveness of the adaptation methodology is demonstrated using both model and flow problems.

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The admission is focused on certain research areas which are mentioned in the links below. If you would like to work with me, check out my website to see what I do and write to me if you have any questions.

For more information see

]]>MS156 - Shear Shallow Water : Modeling and Applications

B. Nkonga (UCA INRIA/CASTOR Nice, France), P. Chandrashekar (TIFR-CAM Bangalore, India), M. Dumbser (Univ. of Trento, Italy) and S. Gavrilyuk (IUSTI Aix Marseille Univ, France)

Shallow flow models provide a far more practical, from the computational standpoint, engineering alternative to the full Euler or Navier-Stokes equations to model free surface flows. This model integrates vertically incompressible flow equations from the topography bed to the flow-free surface (depth integration). Derivations often assume 1) a relatively thin layer flow; 2) minor velocity fluctuations along the flow depth (weakly sheared flow); 3) hydrostatic pressure distribution; 4) a slight bed slope. Depth integration leads to the removal of the need to resolve the free surface explicitly, the reduction of space dimension, and, hopefully, the number of equations to be solved. Despite the reduction and associated simplifications, shallow flow models yield reasonable predictions of some natural process as debris flows, landslides, avalanches, river flows, and even more. However, the classical shallow water model fails in the context of strongly sheared geophysical flows on complex topography. In this context, we must go beyond some modeling assumptions on velocity fluctuation, slopes, and curvature. The usual modeling assumes that the horizontal velocity is weakly varying in the vertical coordinate, which implies that the vertical shear is negligible. The horizontal velocity is the depth average of the three-dimensional velocity field. Since the classical shallow water models assume negligible vertical shear, they cannot model large-scale eddies (rollers) that appear near the surface and behind the hydraulic jump. Under the assumption of the smallness of horizontal vortices, a more general model, the shear shallow water model (SSW), can be derived, including the second-order velocity fluctuation terms. However, the model is principally hyperbolic and nonconservative, posing difficulty in its numerical resolution. We propose to gather active researchers focussing on this model to clarify the situation on different numerical strategies available: advantages and drawbacks. Then we will discuss future directions for investigations.

I last attended ECCOMAS while being a postdoc in 2008, when it was held in Venice, and hope to attend the next one.

]]>Title: Physics-Informed Deep Learning

Ameya D. Jagtap

Division of Applied Mathematics

Brown University, USA

https://appliedmath.brown.edu/people/ameya-jagtap

https://sites.google.com/view/ameyadjagtap18 October 2023

TIFR-CAM, Bangalore

Venue: Ground Floor AuditoriumAbstract: In recent years, physics-informed deep learning (PIDL) has emerged as a powerful tool to solve many problems in the field of computational science. The main idea of PIDL is to incorporate the governing physical laws into a deep learning framework. The PIDL can smoothly integrate the sparse, noisy, and multi-fidelity data along with the governing equations and thereby recast the original PDE problem into an equivalent optimization problem. This approach has various advantages, including the ability to handle ill-posed inverse problems easily. Furthermore, it is a mesh-free approach and is capable of overcoming the curse of dimensionality. In this mini-workshop, I will cover the fundamentals of deep learning as well as physics-informed deep learning through hands-on coding exercises. I will also discuss some advanced PIDL topics, such as its current capabilities, limitations, and various applications, as this is still an active area of research.

- Introduction to Deep Learning and Physics-Informed Deep Learning (10:00 AM - 12:00 PM)
- Performance Improvement Techniques for Physics-Informed Deep Learning (2:00 PM - 4:00 PM)
Each lecture includes hands-on coding exercises. The recommended software:

- TensorFlow 1 or 2 (ML library)
- Python 3.6
- Latex (for plotting figures)

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Boniface will give a talk on 20 September, and Herve’s talk will be announced later.

]]>Approximation of a dispersive model using the Hyperbolic formulation framework: Example of the Conduit equation

B. Nkonga, S. Gavrilyuk, K.M. Shyue

20 September 2023 at 2:00 PM in TIFR-CAM

Recent papers propose hyperbolic regularizations of dispersive equations that are the Euler-Lagrange equations for a given Lagrangian [1]. The primary purpose is to reformulate the high-order derivative dispersive equations as a first-order hyperbolic system. Introducing an appropriate set of parameters gives rise to an extended Lagrangian, asymptotically recovering the proper variational structure of the initial equation under concern. In the case of Serre-Green-Naghdi equations, this “extended Lagrangian” method is now mathematically justified [2]. The advantage of such an approach is obvious: one can use for dispersive equations the entire arsenal of finite volume methods developed for hyperbolic equations.

We aim to apply a similar strategy but in the context of the dispersive conduit equation [3]. According to the classical formulations, this equation does not derive from a Lagrangian, and the classical variational structure is useless. In this context, the purpose is to derive a helpful mathematical pattern for the hyperbolic approximation of the conduit equation.

This presentation will exhibit a mathematical pattern preserved by the hyperbolic reformulation of the conduit equation. Other hyperbolic reformulations exist without fulfilling the proposed mathematical properties. We will compare the numerical solutions based either on hyperbolic or the dispersive formulation of the conduit equation. Contrary to the other hyperbolic models, the “structure-preserving” hyperbolic approximation shows excellent convergence results.

- N. Favrie and S. Gavrilyuk. A rapid numerical method for solving Serre-Green-Naghdi equations describing long free-surface gravity waves. Nonlinearity, 2017.
- V. Duchene. Rigorous justification of the Favrie-Gavrilyuk approximation to the Serre-Green-Naghdi model. Nonlinearity, 2019.
- N.K. Lowman and M.A. Hoefer. Dispersive hydrodynamics in viscous fluid conduits. Phys. Rev. E, 2013.
- S. Gavrilyuk, B. Nkonga, K.M. Shyue, and L. Truskinovsky. Stationary shock-like transition fronts in dispersive systems. Nonlinearity, 2020.

- Arpit Babbar, Praveen Chandrashekar, “Embedded error-based time stepping for Lax-Wendroff Flux Reconstruction for compressible flows”, ICOSAHOM, 14-18 August, 2023, Seoul, Korea.
- Arpit Babbar, Praveen Chandrashekar, Sudarshan Kumar Kenettinkara, “Admissibility preserving subcell limiter for Lax-Wendroff flux reconstruction”, in “MS6 Towards Practical High-Order Methods for Unsteady High-Fidelity Computational Fluid Dynamics”, ICOSAHOM, 14-18 August, 2023, Seoul, Korea.
- Arpit Babbar, Praveen Chandrashekar, Sudarshan Kumar Kenettinkara, “Domain-invariant subcell-based blending limiter for Lax-Wendroff Flux Reconstruction”, ICIAM August 20-25, 2023, Waseda Univ., Tokyo, Japan.

Here he is talking at ICOSAHOM about his work on blending limiter inside Lax-Wendroff Flux Reconstruction scheme; this work is explained here.

More photos are available here.

]]>High Performance Computational Astrophysics on GPUs with Applications

Sethupathy Subramanian

Univ. of Notre Dame, USA1 September 2023 @ 11:00 AM in TIFR-CAM

GPU computing will be an integral part of any Exascale supercomputer in the foreseeable future. It is important to explore how higher-order Godunov (HOG) schemes can be ported to such advanced supercomputer architectures. In this talk, a set of techniques will be identified that enable the efficient and rapid porting of CPU code to GPUs, specifically for HOG schemes used in hyperbolic Partial Differential Equations (PDEs). The focus of this study is on Computational Fluid Dynamics (CFD), Magnetohydrodynamics (MHD), and Computational Electromagnetics (CED), illustrating how HOG schemes can be effectively implemented on supercomputers equipped with GPUs. To express parallelism and optimize code parallelization, the OpenACC language extension is used.

The importance of algorithmic choices that work efficiently within GPU limitations is highlighted, including WENO schemes for spatial reconstruction and ADER predictor steps for space-time representation of PDEs. The implementation examples are presented in both Fortran and C/C++ code using OpenACC extensions, serving as a valuable resource for OpenACC implementations. Challenges related to small GPU caches and optimizing data movement between the host (CPU) and device (GPU) are addressed. The performance of ADER predictor steps and Runge-Kutta time-stepping for CFD and MHD on GPUs is evaluated, showcasing the benefits and limitations of each approach.

The GPU advancements are integrated into the Riemann-Geomesh MHD code, which employs cutting-edge algorithms for simulating spherical astrophysical systems. It utilizes one-sided MPI-3 communication across processors and leverages OpenACC for GPU parallelization. These enhancements enable resource-intensive simulations of magnetically channeled winds from massive stars to be executed and analyzed. The simulations employ high-resolution meshes and strong magnetic fields, necessitating small time steps and long run times on supercomputers. The highly vectorized and GPU-enabled Riemann-Geomesh MHD code achieves approximately 5 times faster execution on GPUs compared to CPUs, with near-perfect scalability up to ~100 GPU processors.