arXiv subjects

Hiroaki Nishikawa

Publications and source records attributed to Hiroaki Nishikawa.

17 recordsLinked to original sources

arXiv2026

On Robust Alpha-Damping Viscous Scheme

In this paper, we investigate the convergence of an implicit defect-correction solver for a viscous discretization based on the alpha-damping scheme for unstructured grids. We show that significantly more robust iterative convergence is achieved by evaluating the damping term at the midpoint between two adjacent cell centers (edge midpoint) rather than at the face centroid. A one-dimensional Fourier analysis reveals that the implicit solver tends to be stable when the damping term in the residual is smaller than that used to construct the Jacobian. This observation suggests that the solver can be stabilized by effectively reducing the magnitude of the damping term in the residual - an effect achieved by the edge-midpoint evaluation. Robust convergence is demonstrated numerically for two-dimensional viscous-flow problems on highly irregular mixed-element and triangular grids.

math.NA↗

arXiv2026

A Flux-Correction Form of the Third-Order Edge-Based Scheme for a General Numerical Flux Function

In this short note, we present a flux-correction form of the third-order edge-based scheme for the Euler equations that enables the direct use of a general flux function. The core idea is to replace, without loss of accuracy, the arithmetic average of the flux extrapolations by a general numerical flux evaluated at the edge midpoint, together with a correction term. We show that the proposed flux-correction form preserves third-order accuracy, provided that the general numerical flux is evaluated with the left and right states that are computed exactly for a quadratic function, which can be achieved effectively by the U-MUSCL scheme with κ = 1/2. Numerical results are presented to verify third-order accuracy with the HLLC and LDFSS flux functions on irregular tetrahedral grids.

math.NA↗

arXiv2026

Edge-based discretizations on triangulations in $\mathbb{R}^d$, with special attention to four-dimensional space

Many time-dependent problems in the field of computational fluid dynamics can be solved using space-time methods. However, such methods can encounter issues with computational cost and robustness. In order to address these issues, efficient, node-centered edge-based schemes are currently being developed. In these schemes, a median-dual tessellation of the space-time domain is constructed based on an initial triangulation. These methods are node-centered or node-based, as the primary components of the discretization are median-dual regions (polytopes) which surround the mesh nodes. These methods are extremely robust, as the median-dual regions will often maintain a positive volume and other good geometric properties, even in cases when some of the associated simplices have negative volumes, or other poor geometric properties. Unfortunately, it is not straightforward to construct median-dual regions or deduce their properties on triangulations for $d \geq 3$. In this work, we provide the first rigorous definition of median-dual regions on triangulations in any number of dimensions. In addition, we introduce a new method for computing the hypervolume of a median-dual region in $\mathbb{R}^d$. Furthermore, we provide a new approach for computing the directed-hyperarea vectors for faces of a median-dual region in $\mathbb{R}^{4}$. These geometric properties are key for developing node-centered edge-based schemes in any number of dimensions. We conclude our work by analyzing the computational complexity of the edge-based schemes, and performing numerical experiments in two, three, and four dimensions. We successfully demonstrate their effectiveness by obtaining accurate solutions to several canonical problems.

math.NA↗

arXiv2025

An Efficient Implementation of Edge-Based Discretization without Forming Dual Control Volumes

This paper shows that lumped directed-area vectors at edges and dual control volumes required to implement the edge-based discretization can be computed without explicitly defining the dual control volume around each node for triangular and tetrahedral grids. It is a simpler implementation because there is no need to form a dual control volume by connecting edge-midpoints, face centroids, and element centroids, and also reduces the time for computing lumped directed-area vectors for a given grid, especially for tetrahedral grids. The speed-up achieved by the proposed algorithm may not be large enough to greatly impact the overall simulation time, but the proposed algorithm is expected to serve as a major stepping stone towards extending the edge-based discretization to four dimensions and beyond (e.g., space-time simulations). Efficient algorithms for computing lumped directed-area vectors and dual volumes without forming dual volumes are presented, and their implementations are described and compared with traditional algorithms in terms of complexity as well as actual computing time for a given grid.

math.NA↗

arXiv2023

On Pitfalls in Accuracy Verification Using Time-Dependent Problems

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 problem is encountered even if the time step is reduced with grid refinement. These can cause a serious problem because then one would wind up trying to find a coding error that does not exist. This short note clarifies the mechanism causing this failure and provides a guide for avoiding such pitfalls

math.NA↗

arXiv2023

An Efficient Quadratic Interpolation Scheme for a Third-Order Cell-Centered Finite-Volume Method on Tetrahedral Grids

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 derivative. It is efficient in that it completely eliminates the need to compute and store second derivatives of solution variables or any other quantities, which are typically required in upgrading a second-order cell-centered unstructured-grid finite-volume discretization to third-order accuracy. Moreover, a high-order flux quadrature formula, as required for third-order accuracy, can also be simplified by utilizing the efficient projected-derivative formula, resulting in a numerical flux at a face centroid plus a curvature correction not involving second derivatives of the flux. Similarly, a source term can be integrated over a cell to high-order in the form of the source term evaluated at the cell centroid plus a curvature correction, again, not requiring second derivatives of the source term. The discretization is defined as an approximation to an integral form of a conservation law but the numerical solution is defined as a point value at a cell center, leading to another feature that there is no need to compute and store geometric moments for a quadratic polynomial to preserve a cell average. Third-order accuracy and improved second-order accuracy are demonstrated and investigated for simple but illustrative test cases in three dimensions.

math.NA↗

arXiv2023

Implicit gradients based conservative numerical scheme for compressible flows

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-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the proposed method's capability using the implicit gradient method for compressible flows.

math.NA↗

arXiv2022

Arithmetic Averages of Viscosity Coefficient are Sufficient for Second-Order Finite-Volume Viscous Discretization on Unstructured Grids

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-exact second-order reconstruction at a face center (e.g., the face center is not located exactly halfway between two adjacent cell centroids) as typical in unstructured grids. Unlike inviscid discretizations, where the solution has to be reconstructed in a linearly exact manner to the face center for second-order accuracy, the viscous discretization does not require the linear exactness for computing viscous coefficients at a face. There are two requirements for second-order accuracy, and the arithmetic average satisfies both of them. Second-order accuracy is numerically demonstrated for a simple one-dimensional nonlinear diffusion problem and for a three-dimensional viscous problem based on methods of manufactured solutions.

math.NA↗

arXiv2021

Economically High-Order Unstructured-Grid Methods: Clarification and Efficient FSR Schemes

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, where highly economical schemes can be constructed based on an extended kappa-scheme combined with economical flux reconstruction formulas, achieving up to fifth-order accuracy (sixth-order with zero dissipation) when a grid is regular. Various economical FSR schemes are presented and their formal orders of accuracy are verified by numerical experiments.

math.NA↗

arXiv2021

On False Accuracy Verification of UMUSCL Scheme

In this paper, we reveal a mechanism behind a false 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. False high-order convergence occurs generally for a scheme that is high order for linear equations but second-order for nonlinear equations. It is caused by unexpected linearization of a target nonlinear equation due to too small of a perturbation added to an exact solution used for accuracy verification. To clarify the mechanism, we begin with a proof that the UMUSCL scheme is third-order accurate only for linear equations. Then, we derive a condition under which the third-order truncation error dominates the second-order error and demonstrate it numerically for Burgers' equation. Similar results are shown for the Euler equations, which disprove some accuracy verification results in the literature. To be genuinely third-order, UMUSCL must be implemented with flux reconstruction.

math.NA↗

arXiv2021

Resolving Confusion Over Third Order Accuracy of U-MUSCL

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=1/2 or kappa=1/3. This paper clarifies some of these confusions: e.g., the U-MUSCL scheme can be third-order accurate in the point-valued solution with kappa=1/3 on regular grids for linear equations in all dimensions, it can be third-order accurate with kappa=1/2 as the QUICK scheme in one dimension. It is shown that the U-MUSCL scheme cannot be third-order accurate for nonlinear equations, except a very special case of kappa=1/2 on regular simplex-element grids, but it can be an accurate low-dissipation second-order scheme. It is also shown that U-MUSCL extrapolates a quadratic function exactly with kappa=1/2 on arbitrary grids provided the gradient is computed by a quadratic least-squares method. Two techniques are discussed, which transform the U-MUSCL scheme into being genuinely third-order accurate on a regular grid: an efficient flux-reconstruction method and a special source term quadrature formula for kappa=1/2.

math.NA↗

arXiv2021

Improved Wall-Normal Derivative Formulae for Anisotropic Adaptive Simplex-Element Grids

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 adaptation, but present a challenge to accurately predict wall-normal derivatives. For example, wall-normal derivatives computed by a simple finite-difference approximation, as typically done in practical fluid-dynamics simulation codes, are often contaminated with numerical noise. To address this issue, we propose an improved method based on a common step-length for the finite-difference approximation, which is otherwise random due to grid irregularity and thus expected to smooth the wall-normal derivative distribution over a boundary. Also, we consider using least-squares gradients to compute the wall-normal derivatives and discuss their possible improvements. Numerical results show that the improved methods greatly reduce the noise in the wall-normal derivatives for irregular simplex-element grids.

math.NA↗

arXiv2020

Grid Quality Measures for Iterative Convergence

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 expected for a grid with a lower value of the F-measure. The G-measure is a least-squares gradient of a specified function around each cell, with the minimum value of zero. Faster convergence is expected for a smaller value of the G-measure towards zero. In this paper, we investigate these measures for inviscid and viscous problems with unstructured grids in two dimensions.

math.NA↗

arXiv2020

On Estimating Machine-Zero Residual

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 solution that is considered as close in magnitude to an exactly-converged solution. One method uses free-stream values as the approximate solution, and the other uses a current solution during an iterative solve as the approximate solution via the method of manufactured solutions. Numerical results show that these estimates predict the levels of machine-zero residuals very accurately for all equations of the Euler and Navier-Stokes equations in a transonic flow over an airfoil and viscous flows over a cylinder and a flat plate.

math.NA↗

arXiv2020

The QUICK Scheme is a Third-Order Finite-Volume Scheme with Point-Valued Numerical Solutions

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 nonlinear conservation law with point-valued solutions stored at cell centers as numerical solutions. Third-order accuracy is proved by a careful and detailed truncation error analysis and demonstrated by a series of thorough numerical tests. The QUICK scheme requires a careful spatial discretization of a time derivative to preserve third-order accuracy for unsteady problems. Two techniques are discussed, including the QUICKEST scheme of Leonard. Discussions are given on how the QUICK scheme is mistakenly found to be second-order accurate. This paper is intended to serve as a reference to clarify any confusion about third-order accuracy of the QUICK scheme and also as the basis for clarifying third-order unstructured-grid schemes as we will discuss in a subsequent paper.

math.NA↗

arXiv2020

A Truncation Error Analysis of Third-Order MUSCL Scheme for Nonlinear Conservation Laws

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 the point value will be strictly made for the numerical solution as well as for the target operator. It is shown that the average of the solutions reconstructed at a face by Van Leer's kappa-scheme recovers a cubic solution exactly with kappa = 1/3, the same is true for the average of the nonlinear fluxes evaluated by the reconstructed solutions, and a dissipation term is already sufficiently small with a third-order truncation error. Finally, noting that the target spatial operator is a cell-averaged flux derivative, we prove that the leading truncation error of the MUSCL finite-volume scheme is third-order with kappa = 1/3. The importance of the diffusion scheme is also discussed: third-order accuracy will be lost when the third-order MUSLC scheme is used with a wrong fourth-order diffusion scheme for convection-diffusion problems. Third-order accuracy is verified by thorough numerical experiments for both steady and unsteady problems. This paper is intended to serve as a reference to clarify confusions about third-order accuracy of the MUSCL scheme, as a guide to correctly analyze and verify the MUSCL scheme for nonlinear equations, and eventually as the basis for clarifying third-order unstructured-grid schemes in a subsequent paper.

physics.comp-ph↗

arXiv2019

First order hyperbolic approach for Anisotropic Diffusion equation

In this paper, we present a high order finite difference solver for anisotropic diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate fifth-order scheme that is independent of the degree of anisotropy is made straightforward by the hyperbolic method with an optimal length scale. We demonstrate that the gradients are computed simultaneously to the same order of accuracy as that of the solution variable by using weight compact finite difference schemes. Furthermore, the approach is extended to improve further the simulation of the magnetized electrons test case previously discussed in Refs.[J. Comput. Phys., 284 (2015) 59-69 and 374 (2018) 1120-1151]. Numerical results indicate that these schemes are capable of delivering high accuracy and the proposed approach is expected to allow the hyperbolic method to be successfully applied to a wide variety of linear and nonlinear problems with anisotropic diffusion.

physics.comp-ph↗