• Title, Summary, Keyword: difference scheme

### A FOURTH-ORDER ACCURATE FINITE DIFFERENCE SCHEME FOR THE EXTENDED-FISHER-KOLMOGOROV EQUATION

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.297-310
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• 2018
• In this paper, a nonlinear high-order difference scheme is proposed to solve the Extended-Fisher-Kolmogorov equation. The existence, uniqueness of difference solution and priori estimates are obtained. Furthermore, the convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete $L^{\infty}-norm$. Some numerical examples are given in order to validate the theoretical results.

### A fourth order finite difference method applied to elastodynamics: Finite element and boundary element formulations

• Souza, L.A.;Carrer, J.A.M.;Martins, C.J.
• Structural Engineering and Mechanics
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• v.17 no.6
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• pp.735-749
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• 2004
• This work presents a direct integration scheme, based on a fourth order finite difference approach, for elastodynamics. The proposed scheme was chosen as an alternative for attenuating the errors due to the use of the central difference method, mainly when the time-step length approaches the critical time-step. In addition to eliminating the spurious numerical oscillations, the fourth order finite difference scheme keeps the advantages of the central difference method: reduced computer storage and no requirement of factorisation of the effective stiffness matrix in the step-by-step solution. A study concerning the stability of the fourth order finite difference scheme is presented. The Finite Element Method and the Boundary Element Method are employed to solve elastodynamic problems. In order to verify the accuracy of the proposed scheme, two examples are presented and discussed at the end of this work.

### AN OVERLAPPING SCHWARZ METHOD FOR SINGULARLY PERTURBED THIRD ORDER CONVECTION-DIFFUSION TYPE

• ROJA, J. CHRISTY;TAMILSELVAN, A.
• Journal of applied mathematics & informatics
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• v.36 no.1_2
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• pp.135-154
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• 2018
• In this paper, an almost second order overlapping Schwarz method for singularly perturbed third order convection-diffusion type problem is constructed. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the combination of classical finite difference scheme and central finite difference scheme on a uniform mesh while on the non-layer region we use the midpoint difference scheme on a uniform mesh. It is shown that the numerical approximations which converge in the maximum norm to the exact solution. We proved that, when appropriate subdomains are used, the method produces convergence of second order. Furthermore, it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantages of this method used with the proposed scheme are it reduce iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.

### A FINITE DIFFERENCE SCHEME FOR RLW-BURGERS EQUATION

• Zhao, Xiaohong;Li, Desheng;Shi, Deming
• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.573-581
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• 2008
• In this paper, a finite difference method for a Cauchy problem of RLW-Burgers equation was considered. Although the equation is not energy conservation, we have given its the energy conservative finite difference scheme with condition. Convergence and stability of the difference solution were proved. Numerical results demonstrate that the method is efficient and reliable.

### Development of a High Accuracy Pure Upwind Difference Scheme (고차 정확도의 순수 상류 차분법의 개발)

• Cho Ji Ryong
• Journal of computational fluids engineering
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• v.4 no.1
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• pp.8-18
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• 1999
• In devising a numerical approximation for the convective spatial transport of a fluid mechanical quantity, it is noted that the convective motion of a scalar quantity occurs in one-way, or from upstream to downstream. This consideration leads to a new scheme termed a pure upwind difference scheme (PUDS) in which an estimated value for a fluid mechanical quantity at a control surface is not influenced from downstream values. The formal accuracy of the proposed scheme is third order accurate. Two typical benchmark problems of a wall-driven fluid flow in a square cavity and a buoyancy-driven natural convection in a tall cavity are computed to evaluate performance of the proposed method. for comparison, the widely used simple upwind scheme, power-law scheme, and QUICK methods are also considered. Computation results are encouraging: the proposed PUDS sensitized to the convection direction produces the least numerical diffusion among tested convection schemes, and, notable improvements in representing recirculation of fluid stream and spatial change of a scalar. Although the formal accuracy of PUDS and QUICK are the same, the accuracy difference of approximately a single order is observed from the revealed results.

### Hybrid-QUICK Scheme Using Finite-Volume Method

• Choi, Jung-Eun
• Journal of Hydrospace Technology
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• v.2 no.2
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• pp.57-67
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• 1996
• The formulation for hybrid-QUICK scheme of convective transport terms in finite-volume calculation procedure is presented. Source terms are modified to apply the hybrid-QUICK scheme. Test calculations are performed for wall-driven cavity flow at Re=$10_2$, $10_3$, and $10_4$. These include the evaluation of boundary conditions approximated by third-order finite difference scheme. The stable and converged solutions are obtained without unsteady terms in the momentum equations. The results using hybrid-QUICK scheme show no difference with those using hybrid scheme at low Re ($=10_2$) and are better at higher Re ($10_3$, and $10_4$).

### Comparison of multigrid performance for higher order scheme with 5-point scheme

• Han, Mun. S.;Kwak, Do Y.;Lee, Jun S.
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.4 no.2
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• pp.135-142
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• 2000
• We consider a multigrid algorithm for higher order finite difference scheme for the Poisson problem on rectangular domain. Several smoothers including Jacobi, Red-black Gauss-Seidel are tested and compared. Since higher order scheme gives much more accurate result then 5-point scheme, one may use small number of levels with higher order scheme and thus the overall cost is reduced quite a lot. The numerical experiment compares the two cases.

### SCHWARZ METHOD FOR SINGULARLY PERTURBED SECOND ORDER CONVECTION-DIFFUSION EQUATIONS

• ROJA, J. CHRISTY;TAMILSELVAN, A.
• Journal of applied mathematics & informatics
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• v.36 no.3_4
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• pp.181-203
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• 2018
• In this paper, we have constructed an overlapping Schwarz method for singularly perturbed second order convection-diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the central finite difference scheme on a uniform mesh while on the non-layer region we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations which converge in the maximum norm to the exact solution. When appropriate subdomains are used, the numerical approximations generated from the method are shown to be first order convergent. Furthermore it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantages of this method used with the proposed scheme is it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.

### Effects of Spatial Discretization Schemes on Numerical Solutions of Viscoelastic Fluid Flows (공간차분도식이 점탄성 유체유동의 수치해에 미치는 영향)

• Min, Tae-Gee;Yoo, Jung-Yul;Choi, Hae-Cheon
• Transactions of the Korean Society of Mechanical Engineers B
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• v.24 no.9
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• pp.1227-1238
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• 2000
• This study examines the effects of the discretization schemes on numerical solutions of viscoelastic fluid flows. For this purpose, a temporally evolving mixing layer, a two-dimensional vortex pair interacting with a wall, and a turbulent channel flow are selected as the test cases. We adopt a fourth-order compact scheme (COM4) for polymeric stress derivatives in the momentum equations. For convective derivatives in the constitutive equations, the first-order upwind difference scheme (UD) and artificial diffusion scheme (AD), which are commonly used in the literature, show most stable and smooth solutions even for highly extensional flows. However, the stress fields are smeared too much and the flow fields are quite different from those obtained by higher-order upwind difference schemes for the same flow parameters. Among higher-order upwind difference schemes, a third-order compact upwind difference scheme (CUD3) shows most stable and accurate solutions. Therefore, a combination of CUD3 for the convective derivatives in the constitutive equations and COM4 for the polymeric stress derivatives in the momentum equations is recommended to be used for numerical simulation of highly extensional flows.

### FINITE DIFFERENCE SCHEME FOR SINGULARLY PERTURBED SYSTEM OF DELAY DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

• SEKAR, E.;TAMILSELVAN, A.
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.3
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• pp.201-215
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• 2018
• In this paper we consider a class of singularly perturbed system of delay differential equations of convection diffusion type with integral boundary conditions. A finite difference scheme on an appropriate piecewise Shishkin type mesh is suggested to solve the problem. We prove that the method is of almost first order convergent. An error estimate is derived in the discrete maximum norm. Numerical experiments support our theoretical results.