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Iterative methods and highorder difference schemes for 2D elliptic problems with mixed derivative
 J. Appl. Math. Comput
"... We consider the twodimensional elliptic equation ..."
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Integrated Fast and High Accuracy Computation of Convection Diffusion Equations Using Multiscale Multigrid Method
, 2009
"... We present an explicit sixth order compact finite difference scheme for fast high accuracy numerical solutions of the two dimensional convection diffusion equation with variable coefficients. The sixth order scheme is based on the wellknown fourth order compact scheme, the Richardson extrapolation ..."
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We present an explicit sixth order compact finite difference scheme for fast high accuracy numerical solutions of the two dimensional convection diffusion equation with variable coefficients. The sixth order scheme is based on the wellknown fourth order compact scheme, the Richardson extrapolation technique, and an operator interpolation scheme. For a particular implementation, we use multiscale multigrid method to compute the fourth order solutions on both the coarse grid and the fine grid. Then an operator interpolation scheme combined with the Richardson extrapolation technique is used to compute a sixth order accurate fine grid solution. We compare the computed accuracy and the implementation cost of the new scheme with the standard ninepoint fourth order compact scheme and SunZhang’s sixth order method. Two convection diffusion problems are solved numerically to validate our proposed sixth order scheme.
ON THE DERIVATION OF HIGHESTORDER COMPACT FINITE DIFFERENCE SCHEMES FOR THE ONE AND TWODIMENSIONAL POISSON EQUATION WITH DIRICHLET BOUNDARY CONDITIONS
"... Abstract. The primary aim of this paper is to answer the question: what are the highestorder five or ninepoint compact finite difference schemes? To answer this question, we present several simple derivations of finite difference schemes for the one and twodimensional Poisson equation on unifor ..."
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Abstract. The primary aim of this paper is to answer the question: what are the highestorder five or ninepoint compact finite difference schemes? To answer this question, we present several simple derivations of finite difference schemes for the one and twodimensional Poisson equation on uniform, quasiuniform, and nonuniform facetoface hyperrectangular grids and directly prove the existence or nonexistence of their highestorder local accuracies. Our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights. For the onedimensional problem, the derivation using the threepoint stencil on both uniform and nonuniform grids yields a scheme with arbitrarily highorder local accuracy. However, for the twodimensional problem, the derivation using the corresponding fivepoint stencil on uniform and quasiuniform grids yields a scheme with at most secondorder local accuracy, and on nonuniform grids yields at most firstorder local accuracy. When expanding the fivepoint stencil to the ninepoint stencil, the derivation using the ninepoint stencil on uniform grids yields at most sixthorder local accuracy, but on quasi and nonuniform grids yields at most fourth and thirdorder local accuracy, respectively.
Open Access Higher Order Compact Scheme Combined with Multigrid Method for Momentum, Pressure Poisson and Energy Equations in Cylindrical Geometry
"... Abstract: A higherorder compact scheme combined with the multigrid method is developed for solving NavierStokes equations along with pressure Poisson and energy equations in cylindrical polar coordinates. The convection terms in the momentum and energy equations are handled in an effective manner ..."
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Abstract: A higherorder compact scheme combined with the multigrid method is developed for solving NavierStokes equations along with pressure Poisson and energy equations in cylindrical polar coordinates. The convection terms in the momentum and energy equations are handled in an effective manner so as to get the fourth order accurate solutions for the flow past a circular cylinder. The superiority of the higher order compact scheme is clearly illustrated in comparison with upwind scheme and defect correction technique by taking a large domain. The developed scheme accurately captures pressure and velocity gradients on the surface when compared to other conventional methods. The pressure in the entire computational domain is computed and the corresponding fourth order accurate pressure fields are plotted. The local Nusselt number and mean Nusselt number are calculated and compared with available experimental and theoretical results.