Hiroaki Nishikawa

National Institute of Aerospace | NIA

In this paper, we propose new Euler flux functions for use in a finite-volume Euler/Navier–Stokes code, which are very simple, carbuncle-free, yet have an excellent boundary-layer-resolving capability, by combining two different Riemann solvers into one based on a rotated Riemann solver approach. We show that very economical Euler flux functions ca...

In this paper, we introduce a general principle for constructing time-accurate diffusion schemes, which is applicable to various discretization methods, including finite-volume, residual-distribution, discontinuous-Galerkin, and spectral-volume methods. The principle is based on a hyperbolic relaxation-system model for diffusion. It is to discretiz...

This paper presents third-order-inviscid implicit edge-based solvers for three-dimensional inviscid and viscous flows on unstructured tetrahedral grids. Third-order edge-based scheme has been implemented into NASA’s FUN3D code for inviscid terms. Second-order edge-based hyperbolic Navier-Stokes schemes, which achieve third-order accuracy in the inv...

This paper discusses the uses of zero and negative volume elements for the node-centered edge-based discretization. It is shown that the edge-based discretization does not suffer from degraded accuracy nor instability with zero-or even negative-volume elements, and that zero and negative-volume elements can be useful for applications such as discon...

We report further progress in the development of agglomerated multigrid techniques for fully unstructured grids in three dimensions. Following the previous studies that identified key elements to grid-independent multigrid convergence for a model equation, and that demon-strated impressive speed-up in single-processor computations for a model diffu...

A novel, efficient, edge-based viscous (EBV) discretization method has been recently developed and implemented in a practical, unstructured-grid, node-centered, finite-volume flow solver. The EBV method is applied to viscous-kernel computations that include evaluations of mean-flow viscous fluxes, turbulence-model and chemistry-model diffusion term...

Want to learn how to write an unstructured CFD code? Grab this code, look inside to see how it is written, get good understanding, and then write your own. This code has Roe and Rotated-RHLL fluxes, Van Albada limiter, and a 2-stage Runge-Kutta time-stepping for solving a shock diffraction problem. It works for quadrilateral grids, triangular grids...

This is a course material for Viscous Fluid Dynamics at Aero&Astro department in Tokai University (Japan) prepared by Prof. Haruo Oguro in 1991 (later translated by Hiroaki Nishikawa). It covers not only viscous flows but also some other topics such as potential flows.

In this short note, we discuss the circumstances that can lead to a failure to observe the design order of discretization error convergence in accuracy verification when solving a time-dependent problem. In particular, we discuss the problem of failing to observe the design order of spatial accuracy with an extremely small time step. The same probl...

Presentation file for AIAA 2023-4419: In this paper, we develop and investigate a time-slab approach to an adaptive-grid space-time hyperbolic Navier-Stokes solver, where a space-time domain spanning over an entire simulation time is sliced into smaller pieces along the time axis. This technique is necessary in order to be able to perform a space-...

This is a presentation file for AIAA 2023-4418: In this paper, we report progress in the development of a third-order accurate, second- derivative-free, shock-capturing finite-volume solver for three-dimensional unstructured grids. The method is economical in the sense that the computation and storage of second derivatives are not required for thi...

Limiters commonly used in the simulations of flows with discontinuities are compared with the new limiter function proposed by Nishikawa using idealized test cases in two dimensions as well as complex three-dimensional problems. The Nishikawa limiter is observed to be consistently the least dissipative in idealized test cases as well as complex pra...

In this paper, we report progress in the development of a third-order accurate, second- derivative-free, shock-capturing finite-volume solver for three-dimensional unstructured grids. The method is economical in the sense that the computation and storage of second derivatives are not required for third-order accuracy. It is based on point-valued nu...

In this paper, we develop and investigate a time-slab approach to an adaptive-grid space-time hyperbolic Navier-Stokes solver, where a space-time domain spanning over an entire simulation time is sliced into smaller pieces along the time axis. This technique is necessary in order to be able to perform a space-time simulation with limited availabili...

In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral grids. The proposed quadratic formula is constructed based on an efficient formula of computing a projected deriva...

In this short note, we discuss the circumstances that can lead to a failure to observe design order of discretization error convergence in accuracy verification when solving a time-dependent problem. Intuitively, one would expect to observe a design spatial order of accuracy when the discretization error is measured on a series of consistently refi...

In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral grids. The proposed quadratic formula is constructed based on an efficient formula of computing a projected deriva...

This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. A problem-in...

Presentation file for AIAA Paper 2023-1605: "Towards High-Order Boundary Procedures for Finite-Volume and Finite-Difference Schemes", presented at AIAA SciTech2023, January 2023

Presentation file for AIAA Paper 2023-1604: "An Algorithm to Detect High-Γ Regions for Three Dimensional Unstructured Grids", presented at AIAA SciTech2023, January 2023.

In this paper, we present high-order boundary procedures for finite-volume and finite difference schemes. The proposed procedures are deliberately constructed with pre-computed derivatives such that they can be implemented without directly accessing cells beyond neighbors and without explicitly setting up ghost cells. Therefore, they can be impleme...

A node-centered, edge-based finite-volume discretization of the compressible Navier-Stokes equations is presented with the heat flux reformulated as a first-order system. A dissipation vector is derived for the reformulated system, such that the heat flux can be upgraded to𝑂 (ℎ3) on simplex element meshes in the same fashion as the inviscid fluxes....

A hypersonic-flow solver has been developed and added to scFLOW, which is a comprehensive computational fluid dynamics (CFD) simulation package designed to handle complex geometries with unstructured grids consisting of arbitrary polyhedral cells and to be user-friendly, allowing non-experts to perform complex simulations relatively easily. The new...

In this paper, we propose an algorithm to estimate a parameter Γ, which is a measure of a combined effect of aspect ratio and curvature associated with a local stencil (e.g., a least-squares stencil), for threedimensional unstructured grids. Stencils with a large Γ are known to cause various troubles in gradient accuracy and iterative convergence o...

Want to learn how to write an implicit unstructured CFD code? Grab this code, look inside to see how it is written, get good understanding, and then write your own. This code computes a steady flow over a bump with the Roe flux by two solution methods: an explicit 2-stage Runge-Kutta scheme and an implicit (defect correction) method with the exact...

This code shows how to compute the source terms in the method of manufactured solutions (MMS) for the 3D Euler equations. Any function can be made an exact solution to the 3D Euler equations with suitable source terms (this code uses an exonential function as an exact solution). MMS has been widely used for verifying the order of accuracy of a CFD...

This is a `unique' book on Computational Fluid Dynamics (CFD). The first half `talks' about governing equations ranging from simple model equations used for algorithm development to the full Euler and Navier-Stokes equations in various forms with complete eigen-structures. Basic materials such as vector identities and coordinate transformations are...

This paper presents a novel, efficient, edge-based viscous (EBV) discretization for finite-volume, node-centered formulations on tetrahedral grids. This new method is implemented in a practical, unstructured-grid Reynolds-averaged Navier–Stokes solver and applied to viscous-kernel computations that include evaluations of mean flow viscous fluxes, t...

In this paper, we present an adaptive-grid space-time solver based on a hyperbolic Navier-Stokes formulation for two-dimensional unsteady viscous flows, where the two-dimensional Navier-Stokes equations are discretized and solved as a steady system in a three-dimensional space-time domain with the coordinates (x, y, t), where t denotes time, using...

This paper establishes consistency and formal second-order accuracy of a novel, efficient, edge-based viscous (EBV) discretization method that has been recently developed, implemented in a practical, unstructured-grid, finite-volume flow solver, and demonstrated multifold acceleration of all viscous-kernel computations that include evaluations of m...

In this paper, we propose a flux correction technique generally applicable to practical finite-volume discretizations of a single flux evaluation per face for achieving second-order accuracy on arbitrary polyhedral grids involving non-planar faces. The proposed technique is derived from the k-exact finite-volume discretization approach originally i...

View Video Presentation: https://doi.org/10.2514/6.2022-4083.vid A novel, efficient, edge-based viscous (EBV) discretization method has been recently developed, implemented in a practical, unstructured-grid, node-centered, finite-volume flow solver, and applied to viscous-kernel computations that include evaluations of meanflow viscous fluxes, turb...

In this paper, we derive analytical formulas for the drag, lift, and moment coefficients of a circular cylinder exposed to a fictitious flow defined by analytical functions in two and three dimensions, and demonstrate that these formulas serve as a useful tool for quickly verifying the implementation of force and moment computation algorithms in co...

In this short note, we discuss the use of arithmetic averages for the evaluation of viscous coefficients such as temperature and velocity components at a face as required in a cell-centered finite-volume viscous discretization on unstructured grids, and show that second-order accuracy can be achieved even when the arithmetic average is not linearly...

Corrections to "New Unstructured-Grid Limiter Functions", AIAA Paper 2022-1374

In this note, I compare the dissipation terms of the viscous upwind fluxes for the hyperbolic Navier-Stokes systems: HNS17 and HNS17G, which are subsystems of HNS20 and HNS20G, respectively. In the hyperbolic Navier-Stoke method, the viscous terms are written as a hyperbolic system and discretized with an upwind flux, resulting in a solver that con...

A space-time edge-based discretization method is developed for solving two-dimensional unsteady viscous-flow problems with adaptive tetrahedral grids in a three-dimensional space-time domain. The method has been implemented in NASA's FUN3D code. The resulting discrete problem is solved by the implicit defect-correction solver with with a multi-colo...

We describe the extension of a 2-D simplified face-averaged nodal-gradient (F-ANG) method to 3-D and demonstrate that the 3-D simplified FANG method is accomplished by augmenting the node-centered gradient least squares stencil. This augmented stencil FANG method is shown to result in advection and diffusion schemes that are stable for hexahedral,...

New Unstructured-Grid Limiter Functions, AIAA SCITECH 2022 Forum, January 2022

In this paper, new unstructured-grid limiter functions are proposed that preserve up to fifth-order accuracy and serve as less-dissipative alternatives to the Venkatakrishnan limiter function for practical unstructured-grid solvers. The new limiters have the following desirable properties: (1) preserving up to fifth-order accuracy in smooth regions...

A sample 3D edge-based Euler code prepared for an educational purpose. It is a serial code and so it can solve only small problems. Also, it runs only on tetrahedral grids. If it helps you understand the algorithms/structure and eventually write your own code, then it will have served its intended purpose. So, please feel free to send me questions...

Explicit iteration towards a steady state by a 1ST-order hyperbolic diffusion scheme and a second-order FD diffusion scheme for solving 0 = nu*uxx + s(x) on an irregularly-spaced grids: 1ST-ORDER accurate solution and gradients by the hyperbolic scheme, 2ND-order accurate solution and 1ST-order accurate gradient by a conventional FD scheme.

Explicit iteration towards a steady state by a 3RD-order hyperbolic diffusion scheme and a 2ND-order FD diffusion scheme (P1 continuous Galerkin finite-element) for solving 0 = nu*uxx + s(x) on an irregularly-spaced grids: 3RD-order accurate solution and gradients by the hyperbolic scheme, 2ND-order accurate solution and 1ST-order accurate gradient...

Explicit iteration towards a steady state by a 2ND-order hyperbolic diffusion scheme and a second-order FD diffusion scheme for solving 0 = nu*uxx + s(x) on an irregularly-spaced grids: 2ND-order accurate solution and gradients by the hyperbolic scheme, 2ND-order accurate solution and 1ST-order accurate gradient by a conventional FD scheme.

In this paper, the authors discuss the unstructured MUSCL (U-MUSCL) reconstruction scheme, an unstructured-grid extension of the 𝜅-scheme of Van Leer, proposed by Burg for the edge-based discretization. This technique has been widely used in practical unstructured-grid fluid-dynamics solvers, but with confusions (e.g., third-order accuracy with 𝜅=1...

This code generates 2D quadrilateral and triangular grids.

Just a program to generate a regular tetrahedral stencil (a stencil at a center node with 14 neighbor nodes with 24 tetrahedra around the center node). Latest version: edu3d_regular_tetra_stencil_v1.zip

This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. The new appr...

高校数学はやはり青春である。それは間違いのない事実である。未だに確信を持てない人は、ぜひ本書を読んで 頂きたい。上巻と同様、本書は形としては高校数学の参考書であるが、その焦点は常に青春という一点に置かれて いる。本書を読み進めるうち、あまりの感動に涙腺がことごとく破壊されるであろう。繰り返す。数学は面白いだとか、数学は楽しいだとか、数学は美しいだとか。そんな主観的なことを言ったとこ ろで、響かない者には響かない。そして、響かせる必要もない。しかし、数学が青春であることは誰もが認識すべ き非常に重要な事実である。特に現役の高校生諸君の場合は、今この時期に理解しておかないと取り返しのつか ないことになる。なぜなら、青春は二度とは戻らないものだからである。本書を通じて、一人でも多くの高校生が その...

高校数学は青春である。それはおそらく間違いのない事実である。そんなことは初耳だという人は、ぜひ本書 を読んで頂きたい。本書は、形としては高校数学の参考書であるが、その焦点は常に青春という一点に置かれてい る。本書を読み進めるうち、あまりの感動に涙が止め処なく溢れ出てくることであろう。数学は面白いだとか、数学は楽しいだとか、数学は美しいだとか。そんな主観的なことを言ったところで響かな い者には響かない。また、響かせる必要もない。しかし、数学が青春であることは誰もが認識すべき非常に重要な 事実である。特に現役の高校生諸君の場合は、今この時期に理解しておかないと取り返しのつかないことになる。 なぜなら、青春は二度とは戻らないものだからである。本書を通じて、一人でも多くの高校生がその青春を力の限 り...

We propose a simplification of the face-averaged nodal-gradient (FANG) method for a cell-centered finite-volume Euler/Navier-Stokes solver on arbitrary grids, and compare it with other gradient methods for trouble-prone grids in two dimensions. The implementa- tion of the FANG method is simplified by adding the face-neighbor cells of the cells arou...

This paper presents an accurate and efficient Poisson solver for wall distance computations on irregular triangular grids. It is discussed that a typical formula of computing the wall distance from a numerical solution of a Poisson equation is not very accurate when solution contours are curved and the accuracy can be improved by a second-order for...

We present a proof by analysis and numerical results that Van Leer's MUSCL conservative scheme with the discretization parameter κ is third-order accurate for κ=1/3. We include both the original finite-volume MUSCL family, updating cell-averaged values of the solution, and the related finite-difference version, updating point values. The presentati...

In this note, I derive the Weiss-Smith preconditioner [1, 2] for the symmetric form of the Euler equations, and show that it is equivalent to Turkel's preconditioner. This is a well known fact.

In this paper, we clarify reconstruction-based discretization schemes for unstructured grids and discuss their economically high-order versions, which can achieve high-order accuracy under certain conditions at little extra cost. The clarification leads to one of the most economical approaches: the flux-and-solution-reconstruction (FSR) approach, w...

A sample MPI code for solving the Laplace equation in a square domain with a regular quad grid

This code solves a 3D linear/nonlinear Poisson equation on an arbitrary tetrahedral or hexahedral grids with Dirichlet/Neumann boundary conditions, by a second-order Jacobian-Free Newton-Krylov edge-based solver. It includes two solvers: (1) a conventional solver with the alpha-damping scheme, and (2) an upwind hyperbolic solver based on the upwind...

In this paper, we reveal a mechanism behind a false accuracy verification encoun- tered with unstructured-grid schemes based on solution reconstruction such as UMUSCL. Third- (or higher-) order of accuracy has been reported for the Euler equations in the liter- ature, but UMUSCL is actually second-order accurate at best for nonlinear equations. Fal...

In this paper, we reveal a mechanism behind a fake accuracy verification encountered with unstructured-grid schemes based on solution reconstruction such as UMUSCL. Third- (or higher-) order of accuracy has been reported for the Euler equations in the literature, but UMUSCL is actually second-order accurate at best for nonlinear equations. Fake hig...

This program computes a machine-zero for a given number by finding the number which does not change the value of the given number when it is added to it. Please read the comments inside the code.

In this note, I describe how a pseudo-time derivative is typically introduced in steady and unsteady solvers, and show that adding a pseudo-time derivative does not destroy time accuracy of an implicit time-stepping scheme nor incur any major additional computational cost. Finally, I discuss some examples of constructing useful discretizations by a...

In this note, I derive BDF2 and BDF3 time-integration schemes for variables step size.

In this note, I derive gradient formulas by weighted least-squares methods on a general non-uniform grid in one dimension. I show that a linear least squares method can be exact for quadratic functions with a particular choice of weights. Accuracy of gradient formulas are numerically verifi�ed for irregular grids.

In this note, I will describe a method that I often use to determine a stretching factor for a desired first spacing or smooth transition from one region to another in my grid generation codes.

In this paper, we discuss the U-MUSCL reconstruction scheme -- an unstructured-grid extension of Van Leer's kappa-scheme -- proposed by Burg for the edge-based discretization [AIAA Paper 2005-4999]. This technique has been widely used in practical unstructured-grid fluid-dynamics solvers but with confusions: e.g., third-order accuracy with kappa=1/...

Just a note on how a one-sided finite-difference formula can be second-order accurate at a boundary, sometimes.

In this paper, we resolve the ever‐present confusion over the QUICK scheme: it is a second‐order scheme or a third‐order scheme. The QUICK scheme, as proposed in the original reference [B. P. Leonard, Comput. Methods. Appl. Mech. Eng., 19, (1979), 59‐98], is a third‐order (not second‐order) finite‐volume scheme for the integral form of a general no...

We present a proof by analysis and numerical results that Van Leer's MUSCL conservative scheme with the discretization parameter κ is third-order accurate for κ = 1/3. We include both the original finite-volume MUSCL family, updating cell-averaged values of the solution, and the related finite-difference version, updating point values. The presenta...

In this paper, we explore methods for computing wall-normal derivatives used for calculating wall skin friction and heat transfer over a solid wall in unstructured simplex-element (triangular/tetrahedral) grids generated by anisotropic grid adaptation. Simplex-element grids are considered as efficient and suitable for automatic grid generation and...

This is an invited paper for the SciTech 2021 special session, High-Fidelity CFD Preworkshop. The paper presents three benchmark cases for verification of Reynolds-averaged Navier-Stokes solvers. The verification studies focus on a one-equation Spalart-Allmaras model, SA-[neg]-QCR2000, that uses a version of quadratic constitutive relations. The be...

In this paper, we discuss the U-MUSCL reconstruction scheme – an unstructured-grid extension of VanLeer’sκ-scheme – proposed by Burg for the edge-based discretization [AIAA Paper 2005-4999]. This technique has been widely used in practical unstructured-grid fluid-dynamics solvers but with confusions: e.g.,third-order accuracy withκ=1/2orκ=1/3. This...

In this short note, we introduce a flexible explicit gradient method for efficient and robust unstructured-grid computations. As shown, the method serves as a memory-efficient alternative to the multiple least-squares gradi- ents needed by conflicting requirements for inviscid, viscous, and turbulence-model source terms (e.g., vorticity), in practi...

In this paper, we clarify reconstruction-based discretization schemes for unstructured grids and discuss their economically high-order versions, which can achieve high-order accuracy under certain conditions at little extra cost. The clarification leads to one of the most economical approaches: the flux-and-solution-reconstruction (FSR) approach, w...

We present an overview of recent developments related to the adaptation of structured and unstructured grids to hypersonic flows, as contained within or utilized by the VULCAN-CFD code. We touch briefly on recent improvements to the relatively mature line-by-line 1-D bow shock and boundary layer edge grid adaptation capability developed to compute...

A new first-order hyperbolic system (FOHS) is formulated for the compressible Navier-Stokes equations. The resulting hyperbolic Navier-Stokes system (HNS), termed HNS20G in this paper, introduces the gradients of density, velocity, and temperature as auxiliary variables. Efficient, accurate, compact and robust reconstructed discontinuous Galerkin (...

This paper is a rebuttal to the claim found in the literature that the MUSCL scheme cannot be third-order accurate for nonlinear conservation laws. We provide a rigorous proof for third-order accuracy of the MUSCL scheme based on a careful and detailed truncation error analysis. Throughout the analysis, the distinction between the cell average and...

This was an internal seminar within NIA.

In this paper, we discuss two grid-quality measures, F- and G-measures, in relation to iterative convergence of an implicit unstructured-grid Navier-Stokes solver. The F-measure is a lower bound of a least-squares gradient, which is a purely geometrical quantity defined in each cell and thus can be computed for a given grid: faster convergence is e...

The alpha-damping scheme is a very popular viscous flux in unstructured-grid codes. This is a basic subroutine that computes the viscous flux with the alpha-damping scheme. It is NOT tested (I only know it complies), but provided here to those who are interested to learn how a viscous flux subroutine is written. This is just an example given for ed...

In this paper, we propose two techniques to estimate the magnitude of a machine-zero residual for a given problem, which is the smallest possible residual that can be achieved when we solve a system of discretized equations. We estimate the magnitude of the machine-zero residual by a norm of residuals computed with a randomly-perturbed approximate...

A sample Matlab code to confirm that the central advection scheme can be stable with the 3rd-order RK time stepping but unstable with 2nd-order RK.

In this paper, we resolve the ever-present confusion over the QUICK scheme: it is a second-order scheme or a third-order scheme. The QUICK scheme, as proposed in the original reference [B. P. Leonard, Comput. Methods. Appl. Mech. Eng., 19, (1979), 59-98], is a third-order (not second-order) finite-volume scheme for the integral form of a general no...

Linearly- and quadratically-exact Implicit edge-based gradient methods are developed for unstructured simplex (triangular and tetrahedral) grids, where gradients are computed, for a given set of function or numerical solution values at nodes, as a solution of a globally-coupled 3�3 block linear system in three dimensions. The method is compact in t...

An upwind space-time edge-based discretization method is developed for solving two-dimensional unsteady inviscid flow problems with adaptive tetrahedral grids in a three-dimensional space-time domain. The resulting discrete problem is solved by a Jacobian-Free Newton-Krylov solver with a multi-color Gauss-Seidel relaxation scheme applied to a� firs...

In this paper, we propose an extension of the face-averaged nodal-gradient cell-centered fnite-volume method to mixed grids, which keeps the residual stencil as small as that of a structured-grid solver and achieves third-order accuracy on uniform quadrilateral grids.The basic idea is to add a jump term to the gradients used for the linear reconstr...

RANS solutions obtained with scFLOW, a polyhedral finite-volume solver developed atSoftware CRADLE, are presented for three-dimensional benchmark problems in NASA’sturbulence model resource website. Grid convergence study is performed, and the resultsare presented and compared with other codes (NASA’s FUN3D, CFL3D, USM3D) for twobenchmark configura...

This paper is a rebuttal to the claim found in the literature that the MUSCL scheme cannot be third-order accurate for nonlinear conservation laws. We provide a rigorous proof for third-order accuracy of the MUSCL scheme based on a careful and detailed truncation error analysis. Throughout the analysis, the distinction between the cell average and...

It is shown that U-MUSCL, an unstructured-grid extension of Van Leer's κ-scheme as proposed by Burg for the edge-based discretization [AIAA Paper 2005-4999], will not be exact for linear functions and thus will not be second-order accurate with a nonzero parameter χ if the flux quadrature point is not exactly halfway between two adjacent solution p...

This is an explicit unstructured-grid Euler code, written for an educational purpose. Read the source code and learn how a cell-centered FV unstructured-grid solver is written. Examples cases are included: transonic airfoil, subsonic cylinder, unsteady vortex, numerical truncation error computation. [The code was used in ODU CFD II, 2018.]

This program tests a linear LSQ method by verifying that it gives the exact gradient for a linear function.

In this paper, a face-averaged nodal-gradient approach is proposed as an efficient gradient method for a second-order cell-centered finite-volume discretization on triangular grids. The gradients needed in the linear reconstruction are computed in two steps: (1) compute gradients at nodes from solutions stored at cells, and (2) compute the gradient...

perform diagonal swapping

This paper presents a robust and efficient Poisson solver that can produce accurate solutions and gradi- ents (e.g., heat flux) on unstructured tetrahedral grids. The solver is constructed based on the hyperbolic method for diffusion, where the Laplacian operator is discretized in the form of a hyperbolic system with solution gradients introduced a...

Simple modification techniques are proposed for making numerical fluxes amenable to unrealizable states (e.g., negative density) without degrading the design order of accuracy, so that a finite-volume solver never fails with unrealizable states arising in the solution reconstruction step and continues to run. The main idea is to evaluate quantities...

This is a presentation file used for the following paper: https://www.researchgate.net/publication/316722881_Accuracy-Preserving_Source_Term_Quadrature_for_Third-Order_Edge-Based_Discretization

This is a 3-minute presentation file for the AIAA paper: A Face-Area-Weighted Centroid Formula for Reducing Grid Skewness and Improving Convergence of Edge-Based Solver on Highly-Skewed Simplex Grids.