CONVERGENCE ANALYSIS ON GIBOU - MIN METHOD FOR THE SCALAR FIELD IN HODGE - HELMHOLTZ DECOMPOSITION

CHOHONG MIN,GANGJOON YOON 한국산업응용수학회 2014 Journal of the Korean Society for Industrial and A Vol.18 No.4

The Hodge-Helmholtz decomposition splits a vector field into the unique sum of a divergence-free vector field (solenoidal part) and a gradient field (irrotational part). In a bounded domain, a boundary condition needs to be supplied to the decomposition. The decomposition with the non-penetration boundary condition is equivalent to solving the Poisson equation with the Neumann boundary condition. The Gibou-Min method is an application of the Poisson solver by Purvis and Burkhalter to the decomposition. Using the L<SUP>2</SUP>-orthogonality between the error vector and the consistency, the convergence for approximating the divergence-free vector field was recently proved to be O(h<SUP>1.5</SUP>) with step size h: In this work, we analyze the convergence of the irrotattional in the decomposition. To the end, we introduce a discrete version of the Poincare inequality, which leads to a proof of the O(h) convergence for the scalar variable of the gradient field in a domain with general intersection property.

A Heaviside-function approach for simulating fluid-structure interaction

Chohong Min,Frederic Gibou 한국산업응용수학회 2012 한국산업응용수학회 학술대회 논문집 Vol.7 No.1

A SECOND ORDER ACCURATE LEVEL SET METHOD ON QUADTREE GRIDS

Chohong Min,Frederic Gibou 한국산업응용수학회 2009 한국산업응용수학회 학술대회 논문집 Vol.2009 No.5

We present a level set method on non-graded adaptive Cartesian grids, i.e. grids for which the ratio between adjacent cells is not constrained. We use quadtree and octree data structures to represent the grid and a simple algorithm to generate a mesh with the finest resolution at the interface. In particular, we present (1) a locally third order accurate reinitialization scheme that transforms an arbitrary level set function into a signed distance function, (2) a second order accurate semi-Lagrangian methods to evolve the linear level set advection equation under an externally generated velocity field, (3) a second order accurate upwind method to evolve the non-linear level set equation under a normal velocity as well as to extrapolate scalar quantities across an interface in the normal direction, and (4) a semi-implicit scheme to evolve the interface under mean curvature. Combined, we obtain a level set method on adaptive Cartesian grids with a negligible amount of mass loss. We propose numerical examples in two and three spatial dimensions to demonstrate the accuracy of the method.

A NEW UNDERSTANDING OF THE QR METHOD

CHOHONG MIN 한국산업응용수학회 2010 Journal of the Korean Society for Industrial and A Vol.14 No.1

The QR method is one of the most common methods for calculating the eigenvalues of a square matrix, however its understanding would require complicated and sophisticated mathematical logics. In this article, we present a simple way to understand QR method only with a minimal mathematical knowledge. A deflation technique is introduced, and its combination with the power iteration leads to extracting all the eigenvectors. The orthogonal iteration is then shown to be compatible with the combination of deflation and power iteration. The connection of QR method to orthogonal iteration is then briefly reviewed. Our presentation is unique and easy to understand among many accounts for the QR method by introducing the orthogonal iteration in terms of deflation and power iteration.

An energy-stable method for solving the incompressible Navier–Stokes equations with non-slip boundary condition

Lee, Byungjoon,Min, Chohong Elsevier 2018 Journal of computational physics Vol.360 No.-

<P><B>Abstract</B></P> <P>We introduce a stable method for solving the incompressible Navier–Stokes equations with variable density and viscosity. Our method is stable in the sense that it does not increase the total energy of dynamics that is the sum of kinetic energy and potential energy. Instead of velocity, a new state variable is taken so that the kinetic energy is formulated by the <SUP> L 2 </SUP> norm of the new variable. Navier–Stokes equations are rephrased with respect to the new variable, and a stable time discretization for the rephrased equations is presented.</P> <P>Taking into consideration the incompressibility in the Marker-And-Cell (MAC) grid, we present a modified Lax–Friedrich method that is <SUP> L 2 </SUP> stable. Utilizing the discrete integration-by-parts in MAC grid and the modified Lax–Friedrich method, the time discretization is fully discretized. An explicit CFL condition for the stability of the full discretization is given and mathematically proved.</P> <P><B>Highlights</B></P> <P> <UL> <LI> We present a modified Lax–Friedrich method that is <SUP> L 2 </SUP> stable. </LI> <LI> A novel time discretization for the Navier–Stokes' equations are presented. </LI> <LI> An explicit CFL condition for the stability of the full discretization is given and mathematically proved. </LI> <LI> We introduce a stable method for solving the incompressible Navier–Stokes' equations with variable density and viscosity. </LI> </UL> </P>

SURFACE RECONSTRUCTION FROM SCATTERED POINT DATA ON OCTREE

CHANGSOO PARK,CHOHONG MIN,MYUNGJOO KANG 한국산업응용수학회 2012 Journal of the Korean Society for Industrial and A Vol.16 No.1

In this paper, we propose a very efficient method which reconstructs the high resolution surface from a set of unorganized points. Our method is based on the level set method using adaptive octree. We start with the surface reconstruction model proposed in [20]. In [20], they introduced a very fast and efficient method which is different from the previous methods using the level set method. Most existing methods[21, 22] employed the time evolving process from an initial surface to point cloud. But in [20], they considered the surface reconstruction process as an elliptic problem in the narrow band including point cloud. So they could obtain very speedy method because they didn’t have to limit the time evolution step by the finite speed of propagation. However, they implemented that model just on the uniform grid. So they still have the weakness that it needs so much memories because of being fulfilled only on the uniform grid. Their algorithm basically solves a large linear system of which size is the same as the number of the grid in a narrow band. Besides, it is not easy to make the width of band narrow enough since the decision of band width depends on the distribution of point data. After all, as far as it is implemented on the uniform grid, it is almost impossible to generate the surface on the high resolution because the memory requirement increases geometrically. We resolve it by adapting octree data structure[12, 11] to our problem and by introducing a new redistancing algorithm which is different from the existing one[19].

AN ELEMENTARY PROOF OF THE OPTIMAL RECOVERY OF THE THIN PLATE SPLINE RADIAL BASIS FUNCTION

MORAN KIM,CHOHONG MIN 한국산업응용수학회 2015 Journal of the Korean Society for Industrial and A Vol.19 No.4

In many practical applications, we face the problem of reconstruction of an unknown function sampled at some data points. Among infinitely many possible reconstructions, the thin plate spline interpolation is known to be the least oscillatory one in the Beppo-Levi semi norm, when the data points are sampled in R<SUP>2</SUP>. The traditional proofs supporting the argument are quite lengthy and complicated, keeping students and researchers off its understanding. In this article, we introduce a simple and short proof for the optimal reconstruction. Our proof is unique and reguires only elementary mathematical background.

Comparison of eigenvalue ratios in artificial boundary perturbation and Jacobi preconditioning for solving Poisson equation

Yoon, Gangjoon,Min, Chohong Elsevier 2017 Journal of computational physics Vol.349 No.-

<P><B>Abstract</B></P> <P>The Shortley–Weller method is a standard finite difference method for solving the Poisson equation with Dirichlet boundary condition. Unless the domain is rectangular, the method meets an inevitable problem that some of the neighboring nodes may be outside the domain. In this case, an usual treatment is to extrapolate the function values at outside nodes by quadratic polynomial. The extrapolation may become unstable in the sense that some of the extrapolation coefficients increase rapidly when the grid nodes are getting closer to the boundary. A practical remedy, which we call artificial perturbation, is to treat grid nodes very near the boundary as boundary points. The aim of this paper is to reveal the adverse effects of the artificial perturbation on solving the linear system and the convergence of the solution. We show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues is an important factor in solving the linear system. Our analysis shows that the artificial perturbation results in a small enhancement of the eigenvalue ratio from O ( 1 / ( h ⋅ <SUB> h m i n </SUB> ) to O ( <SUP> h − 3 </SUP> ) and triggers an oscillatory order of convergence. Instead, we suggest using Jacobi or ILU-type preconditioner on the matrix without applying the artificial perturbation. According to our analysis, the preconditioning not only reduces the eigenvalue ratio from O ( 1 / ( h ⋅ <SUB> h m i n </SUB> ) to O ( <SUP> h − 2 </SUP> ) , but also keeps the sharp second order convergence.</P>

A Stable and Convergent Hodge Decomposition Method for Fluid-Solid Interaction

Yoon, Gangjoon,Min, Chohong,Kim, Seick Springer-Verlag 2018 Journal of scientific computing Vol.76 No.2

A REVIEW OF THE SUPRA-CONVERGENCES OF SHORTLEY-WELLER METHOD FOR POISSON EQUATION

GANGJOON YOON,CHOHONG MIN 한국산업응용수학회 2014 Journal of the Korean Society for Industrial and A Vol.18 No.1

The Shortley-Weller method is a basic finite difference method for solving the Poisson equation with Dirichlet boundary condition. In this article, we review the analysis for supra-convergence of the Shortley-Weller method. Though consistency error is first order accurate at some locations, the convergence order is globally second order. We call this increase of the order of accuracy, supra-convergence. Our review is not a simple copy but serves a basic foundation to go toward yet undiscovered analysis for another supra-convergence: we present a partial result for supra-convergence for the gradient of solution.