Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 57
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(10.91), find foreach of the cases the error matrix defined as R = M − P.Exercise 8Solve the following systems of equations using the ILU(0) and DILU methods.Perform three iterations only starting with a zero initial guess.10.7Exercises3610ðaÞ1B5BB3A¼BB4B@430ðbÞ12B4BB4A¼BB3B@30352123007202000700001042102 CC2 CC;1 CC0 A10375200117020400610430270110 CC0 CC;0 CC1 A1126B 28 CB CB 18 CCb¼BB 26 CB C@ 11 A260125B 24 CB CB4 CCb¼BB4 CB C@2 A250Exercise 9Solve the systems of equations in Exercise 8 using the Gauss-Seidel and Jacobimethods (starting with a zero initial guess) and compare the errors after threeiterations with the errors obtained in exercise 8 and with the exact solution.Comment on the results.Exercise 10Solve the following symmetric systems of equations by performing two iterationsof the preconditioned conjugate gradient method (start with a zero field):0ðaÞ6B0BA¼BB 1@01061110ðbÞ4B1BA¼BB0@ 100ðcÞ3B1BB 1A¼BB2B@ 101711221160001041111 CC0 CC;1 A60250011060103CC0CC;0A515213117000210602120061017B9 CB CCb¼BB 27 C@ 23 A1018B2 CB CCb¼BB 12 C@ 11 A30102 CC0 CC;2 CC1 A7129B9 CB CB3 CCb¼BB 13 CB C@ 28 A27036210Solving the System of Algebraic EquationsExercise 11List all the available linear solvers inside OpenFOAM®.After choosing the smoothSolver in OpenFOAM®, list all implemented smoothers.Exercise 12Find in OpenFOAM® the implementation of the BiCG linear solver (PBiCG) andcompare it with the BiCG algorithm of Lanczos.Exercise 13Find in OpenFOAM® the implementation of the multigrid V-cycle ($FOAM_SRC/OpenFOAM/matrices/lduMatrix/solvers/GAMG/GAMGSolverSolve.C) and compare it with the theoretical V-cycle algorithm.Exercise 14Verify the correct implementation in OpenFOAM® of the diagonal version of theILU(0) developed by Pommerell.References1.
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In: McCormick SF (ed) Multigridmethods, frontiers in applied mathematics, vol 3. SIAM, Philadelphia, pp 73–13051. OpenFOAM, 2015 Version 2.3.x. http://www.openfoam.orgChapter 11Discretization of the Convection TermAbstract So far, the discretization of the general steady diffusion equation hasbeen formulated on orthogonal, non-orthogonal, structured, and unstructured grids.Another important term, the convection term represented by the divergence operator, is the focus of this chapter. Initially this term is discretized using a symmetricallinear profile similar to the one adopted for the discretization of the diffusion term.The shortcomings of this profile are delineated and a remedy is suggested throughthe use of an upwind profile.
Even though it leads to physically plausible predictions, the upwind profile is shown to be highly diffusive generating results that arefirst order accurate. To increase accuracy, higher order profiles that are upwindbiased are introduced. While reducing the discretization error, higher order profilesare shown to give rise to another type of error known as the dispersion error.Methods dealing with this error will be dealt with in the next chapter. Moreover, theflow field, which represents the driving catalyst of the convection term, is assumedto be known.
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