arXiv subjects

Suyash Shrestha

Publications and source records attributed to Suyash Shrestha.

6 recordsLinked to original sources

arXiv2026

An Approximate Bounded Cochain Projection

This paper presents a construction of a projector from an infinite-dimensional Hilbert complex of differential $k$-forms onto a finite-dimensional piecewise polynomial sub-complex. We demonstrate that, on contractable domains, the proposed projector attains the three properties of the Bounded Cochain Projector, namely that the projector is idempotent, uniformly bounded in the Sobolev norm, and it commutes with the exterior derivative. On non-contractible domains, the projector remains idempotent and uniformly bounded, while the commuting property is satisfied up to arbitrary accuracy.

math.NA↗

arXiv2026

Projection-Based Reconstruction for Achieving High-Order Accuracy from Low-Order DGSEM Simulations

High-order discontinuous Galerkin spectral element methods (DGSEM) based on Legendre-Gauss-Lobatto (LGL) nodes provide accurate and efficient discretizations for conservation laws. However, their cost, memory footprint, and time-step restrictions increase rapidly when the degree of the polynomial increases. This paper develops a corrected $\mathbb{P}_n\mathbb{P}_m$ ($c\mathbb{P}_n\mathbb{P}_m$) approach for DGSEM-LGL discretizations that aims to recover the accuracy of an $m^{th}$-order approximation while evolving only the degrees of freedom associated with an $n^{th}$-order representation, with $n<m$. The projected evolution of the high-order components is derived first at the continuous level and then in the fully discrete DGSEM-LGL setting. The discrete analysis shows that because LGL quadrature is not exact for the highest Legendre mode, a correction term for that mode is required to preserve the order of convergence. A compact projection-based reconstruction operator is then introduced to recover high-order components without solving the enlarged constrained least-squares systems used in standard reconstruction procedures. For sufficiently smooth solutions, the resulting $c\mathbb{P}_n\mathbb{P}_m$ scheme is shown to achieve the expected $m+1^{th}$ convergence order. Numerical experiments for one- and two-dimensional conservation laws, including Euler, viscous Burgers, and 2D decaying homogeneous isotropic turbulence, confirm theoretical convergence behavior and demonstrate competitive accuracy relative to computational cost, with particularly clear efficiency gains for viscous flows.

math.NA↗

arXiv2025

Optimal solutions employing an algebraic Variational Multiscale approach Part II: Application to Navier-Stokes

This work presents a non-linear extension of the high-order discretisation framework based on the Variational Multiscale (VMS) method previously introduced for steady linear problems. We build on the concept of an optimal projector defined via the symmetric part of the governing operator. Using this idea, we generalise the formulation to the two-dimensional incompressible Navier-Stokes equations. The approach maintains a clear separation between resolved and unresolved scales, with the fine-scale contribution approximated through the approximate Fine-Scale Greens' operator of the associated symmetric operator. This enables a consistent variational treatment of non-linearity while preserving high-order accuracy. We show that the method yields numerical solutions that closely approximate the optimal projection of the continuous/highly-resolved solution and inherits desirable conservation properties. Particularly, the formulation guarantees discrete conservation of mass, energy, and vorticity, where enstrophy conservation is also achieved when exact or over-integration is employed. Numerical results confirm the methodology's robustness and accuracy, while also demonstrating its computational cost advantage compared to the baseline Galerkin approach for the same accuracy.

math.NA↗

arXiv2025

Optimal solutions employing an algebraic Variational Multiscale approach Part I: Steady Linear Problems

This work extends our previous study from S. Shrestha et al. (2024) by introducing a new abstract framework for Variational Multiscale (VMS) methods at the discrete level. We introduce the concept of what we define as the optimal projector and present a discretisation approach that yields a numerical solution closely approximating the optimal projection of the infinite-dimensional continuous solution. In this approach, the infinite-dimensional unresolved scales are approximated in a finite-dimensional subspace using the numerically computed Fine-Scale Greens' function of the underlying symmetric problem. The proposed approach involves solving the VMS problem on two separate meshes: a coarse mesh for the full PDE and a fine mesh for the symmetric part of the continuous differential operator. We consider the 1D and 2D steady advection-diffusion problems in both direct and mixed formulations as the test cases in this paper. We first present an error analysis of the proposed approach and show that the projected solution is achieved as the approximate Greens' function converges to the exact one. Subsequently, we demonstrate the working of this method where we show that it can exponentially converge to the chosen optimal projection. We note that the implementation of the present work employs the Mimetic Spectral Element Method (MSEM), although, it may be applied to other Finite/Spectral Element or Isogeometric frameworks. Furthermore, we propose that VMS should not be viewed as a stabilisation technique; instead, the base scheme should be inherently stable, with VMS enhancing the solution quality by supplementing the base scheme.

math.NA↗

arXiv2025

Implementation of Immersed Boundaries via Volume Penalization in the Industrial Aeronautical Computational Fluid Dynamics Solver CODA

We present the implementation and validation of an immersed boundary volume penalization method in the computational fluid dynamics solver CODA (from ONERA, DLR, and Airbus). Our goal is to model and simulate turbulent fluid flows in complex 3D aerodynamic configurations through the numerical solution of the Reynolds--averaged Navier--Stokes equations using the Spalart--Allmaras turbulent model. To do that, an immersed boundary method has been implemented in CODA and an efficient preprocessing tool for the construction of unstructured hexahedral meshes with adaptive mesh refinement around immersed geometries has been developed. We report several numerical examples, including subsonic flow past the NACA0012 airfoil, transonic flow past the RAE2822 airfoil, subsonic flow past the MDA30P30N multi-element airfoil, and subsonic flow around the NASA high-lift CRM aircraft. These simulations have been performed in the CODA solver with a second-order finite volume scheme as spatial discretization and an implicit backward Euler scheme based on the matrix-free GMRES block-Jacobi iterative method. The reported numerical simulations are in good agreement with their corresponding experimental data. These encouraging results allow us to conclude that the implemented immersed boundary method is efficient, flexible, and accurate and can therefore be used for aeronautical applications in industry.

physics.flu-dyn↗

arXiv2024

Construction and application of an algebraic dual basis and the Fine-Scale Greens' Function for computing projections and reconstructing unresolved scales

In this paper, we build on the work of [T. Hughes, G. Sangalli, VARIATIONAL MULTISCALE ANALYSIS: THE FINE-SCALE GREENS' FUNCTION, PROJECTION, OPTIMIZATION, LOCALIZATION, AND STABILIZED METHODS, SIAM Journal of Numerical Analysis, 45(2), 2007] dealing with the explicit computation of the Fine-Scale Green's function. The original approach chooses a set of functionals associated with a projector to compute the Fine-Scale Green's function. The construction of these functionals, however, does not generalise to arbitrary projections, higher dimensions, or Spectral Element methods. We propose to generalise the construction of the required functionals by using dual functions. These dual functions can be directly derived from the chosen projector and are explicitly computable. We show how to find the dual functions for both the $L^2$ and the $H^1_0$ projections. We then go on to demonstrate that the Fine-Scale Green's functions constructed with the dual basis functions consistently reproduce the unresolved scales removed by the projector. The methodology is tested using one-dimensional Poisson and advection-diffusion problems, as well as a two-dimensional Poisson problem. We present the computed components of the Fine-Scale Green's function, and the Fine-Scale Green's function itself. These results show that the method works for arbitrary projections, in arbitrary dimensions. Moreover, the methodology can be applied to any Finite/Spectral Element or Isogeometric framework.

math.NA↗