The rotational incremental pressure-correction (RIPC) scheme, described in Timmermans et al. [Int... more The rotational incremental pressure-correction (RIPC) scheme, described in Timmermans et al. [Int. J. Numer. Methods. Fluids., 22 (1996)] and Shen et al. [Math. Comput., 73 (2003)] for non-rotational Navier-Stokes equations, is extended to rotating incompressible flows. The method is implemented in the context of a pseudo Fourier-spectral code and applied to several rotating laminar and turbulent flows. The performance of the scheme and the computational results are compared to the so-called diagonalization method (DM) developed by Morinishi et al. [Int. J. Heat. Fluid. Flow., 22 (2001)]. The RIPC predictions are in excellent agreement with the DM predictions, while being simpler to implement and computationally more efficient. The RIPC scheme is not in anyway limited to implementation in a pseudo-spectral code or periodic boundary conditions, and can be used in complex geometries and with other suitable boundary conditions.
The purpose of this note is to correct some errors in the proofs of pressure error estimates in a... more The purpose of this note is to correct some errors in the proofs of pressure error estimates in and . The origin of the errors is the following incorrect inequality 1 (1) ), which the author incorrectly derived by identifying u V with u -1 for u ā H. 2 The correct inequality is: (2) However, the incorrect proofs induced by the error (1) can all be fixed as indicated below. More precisely, the proofs for the velocity error estimates in [1, 2] are all valid provided some minor changes of notations; the pressure error estimates still hold, but their proofs necessitate some additional estimates. In summary, all the results presented in remain valid provided some additional regularity assumptions on the exact solution are made. In the following we provide the details for the aforementioned modifications and corrections. This work was supported in part by NSF grant DMS-9205300 1 The incorrectness of (1) was pointed out to me by R. Temam to whom I am grateful. Later on J.L. Guermond has also expressed to me his doubts about (1) 2 The confusion can be partly explained by a technical sublety. We know that V ā H 1 0 (ā¦) d and hence H 1 0 (ā¦) d = V ā V ā„ . By the Riesz theorem and elementary properties of Hilbert spaces, V is isomorphic to and could be identified with a subspace of H -1 (ā¦) d . However this identification must not be made because, as it is usual with evolution equations, we already identified H with a subspace of V (V ā H ā V ). Hence this double identification is not allowed
We developed a numerical method for computing the effective properties of a microstructure. The m... more We developed a numerical method for computing the effective properties of a microstructure. The method is particularly efficient and accurate for microstructures with a diffuse-interface description similar to those generated from phase-field simulations. In particular, we considered the diffusive transport property of a microstructure by solving the steady-state diffusion equation using a Fourier-Chebyshev spectral method. Computed effective diffusivities agree very well with existing analytical solutions and computer simulations for a number of simple model systems. Combining with the phase-field model for simulating microstructure evolution, the proposed method can be applied for modeling the temporal evolution of effective properties. This is illustrated by considering grain growth and the corresponding effective transport property evolution as function of time.
A numerical simulation of the unsteady incompressible flow in the unit cavity is performed by usi... more A numerical simulation of the unsteady incompressible flow in the unit cavity is performed by using a Chebyshev-Tau approximation for the space variables. The high accuracy of the spectral methods and the condensed distribution of the Chebyshev-collocation points near the boundary enable us to obtain reliable results for high Reynolds numbers with a moderate number of modes. It is found that the flow converges to a stationary state for Reynolds numbers (Re) up to 10,000; for Reynolds numbers larger than a critical value 10,000 < Re 1 < 10,500 and less than another critical value 15,000 < Re 2 < 15,500, the flow becomes periodic in time which indicates a Hoph bifurcation; the flow loses time periodicity for Re > Re 2.
Two new Gauge-Uzawa schemes are constructed for incompressible flows with variable density. One i... more Two new Gauge-Uzawa schemes are constructed for incompressible flows with variable density. One is in the conserved form while the other is in the convective form. It is shown that the first-order versions of both schemes, in their semi-discretized form, are unconditionally stable. Numerical experiments indicate that the first-order (resp. second-order) versions of the two schemes lead to first-order (resp. second-order) convergence rate for all variables and that these schemes are suitable for handling problems with large density ratios such as in the situation of air bubble rising in water.
International Journal for Numerical Methods in Fluids, 2006
In Gresho and Sani (Int. J. Numer. Methods Fluids 1987; 7:1111ā1145; Incompressible Flow and the ... more In Gresho and Sani (Int. J. Numer. Methods Fluids 1987; 7:1111ā1145; Incompressible Flow and the Finite Element Method, vol. 2. Wiley: New York, 2000) was proposed an important hypothesis regarding the pressure Poisson equation (PPE) for incompressible flow: Stated there but not proven was a soācalled equivalence theorem (assertion) that stated/asserted that if the NavierāStokes momentum equation is solved simultaneously with the PPE whose boundary condition (BC) is the Neumann condition obtained by applying the normal component of the momentum equation on the boundary on which the normal component of velocity is specified as a Dirichlet BC, the solution (u, p) would be exactly the same as if the āprimitiveā equations, in which the PPE plus Neumann BC is replaced by the usual divergenceāfree constraint (ā Ā· u = 0), were solved instead.This issue is explored in sufficient detail in this paper so as to actually prove the theorem for at least some situations. Additionally, like the ori...
A simple and eective numerical method is proposed for simulating the temporal diusive mass transp... more A simple and eective numerical method is proposed for simulating the temporal diusive mass transport process through a microstructure with arbitrary complexity described by a phase-Ā®eld approach. The mass diusion through a given microstructure is modeled by a diusion equation with a variable diusion coecient, which is solved by an ecient and accurate semi-implicit spectral method. It is shown that it is possible to extract the eective diusion coecient for any given microstructure from the temporal concentration proĀ®les. The method is used to simulate the grain boundary diusion in a single-phase polycrystalline grain structure and the heterogeneous diusion in a twophase microstructure with dierent diusion coecient in each phase. Results are compared with existing analytical theories and computer simulations.
Journal of Computational and Applied Mathematics, 2014
This paper is concerned with approximation properties of orthonormal mapped Chebyshev functions (... more This paper is concerned with approximation properties of orthonormal mapped Chebyshev functions (OMCFs) in unbounded domains. Unlike the usual mapped Chebyshev functions which are associated with weighted Sobolev spaces, the OMCFs are associated with the usual (non-weighted) Sobolev spaces. This leads to particularly simple stiffness and mass matrices for higher-dimensional problems. The approximation results for both the usual tensor product space and hyperbolic cross space are established, with the latter particularly suitable for higher-dimensional problems.
The incompressible Stokes equations with prescribed normal stress (open) boundary conditions on p... more The incompressible Stokes equations with prescribed normal stress (open) boundary conditions on part of the boundary are considered. It is shown that the standard pressure-correction method is not suitable for approximating the Stokes equations with open boundary conditions, whereas the rotational pressure-correction method yields reasonably good error estimates. These results appear to be the first ever published for splitting schemes with open boundary conditions. Numerical results in agreement with the error estimates are presented.
A fast Chebyshev-Fourier algorithm for Poisson-type equations in polar geometries is presented in... more A fast Chebyshev-Fourier algorithm for Poisson-type equations in polar geometries is presented in this paper. The new algorithm improves upon the algorithm of Jie Shen (1997), by taking advantage of the odd-even parity of the Fourier expansion in the azimuthal direction, and it is shown to be more efficient in terms of CPU and memory.
Boundary perturbation methods, in which the deviation of the problem geometry from a simple one i... more Boundary perturbation methods, in which the deviation of the problem geometry from a simple one is taken as the small quantity, have received considerable attention in recent years due to an enhanced understanding of their convergence properties. One approach to deriving numerical methods based upon these ideas leads to Bruno and Reitich's generalization [Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), pp. 317-340] of Rayleigh and Rice's classical algorithm giving the "method of variation of boundaries" which is very fast and accurate within its domain of applicability. Treating problems outside this domain (e.g., boundary perturbations which are large and/or rough) led Nicholls and Reitich [Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), pp. 1411-1433] to design the "transformed field expansions" (TFE) method, and the rigorous numerical analysis of these recursions is the subject of the current work. This analysis is based upon analyticity estimates for the TFE expansions coupled to the convergence of Fourier-Legendre Galerkin methods. This powerful and flexible analysis is extended to a wide range of problems including those governed by Laplace and Helmholtz equations, and the equations of traveling free-surface ideal fluid flow.
We propose a new weighted weak formulation for the Fokker-Planck equation of the finitely extensi... more We propose a new weighted weak formulation for the Fokker-Planck equation of the finitely extensible nonlinear elastic dumbbell model and prove its well-posedness in weighted Sobolev spaces. We also propose simple and efficient semi-implicit time-discretization schemes which are unconditionally stable, i.e., the step size of time marching does not depend on the number of the bases used in configurational space. We then restrict ourselves to the two-dimensional case and construct two Fourier-Jacobi spectral-Galerkin algorithms which enjoy the following properties: (i) they are unconditionally stable, spectrally accurate, and of optimal computational complexity; (ii) they conserve the volume and provide accurate approximation to higher-order moments of the distribution function; and (iii) they can be easily extended to coupled nonhomogeneous systems. Numerical results are presented to show how to choose a proper weight to get the best numerical results of the distribution function and the polymer stress.
Mathematical modeling and numerical simulation of smectic C liquid crystals which possess the spo... more Mathematical modeling and numerical simulation of smectic C liquid crystals which possess the spontaneous polarization are considered in this paper. In particular, the model allows for a system with a zero net polarization which is one of the ubiquitous systems of the polarized liquid crystals. Theoretical and numerical investigations are conducted to study effects of the energy associated with the polarization, switching patterns between two uniform states by an externally applied field and random noise, as well as a relation between polarization and applied field near the phase transition from the smectic A and smectic C.
The rotational incremental pressure-correction (RIPC) scheme, described in Timmermans et al. [Int... more The rotational incremental pressure-correction (RIPC) scheme, described in Timmermans et al. [Int. J. Numer. Methods. Fluids., 22 (1996)] and Shen et al. [Math. Comput., 73 (2003)] for non-rotational Navier-Stokes equations, is extended to rotating incompressible flows. The method is implemented in the context of a pseudo Fourier-spectral code and applied to several rotating laminar and turbulent flows. The performance of the scheme and the computational results are compared to the so-called diagonalization method (DM) developed by Morinishi et al. [Int. J. Heat. Fluid. Flow., 22 (2001)]. The RIPC predictions are in excellent agreement with the DM predictions, while being simpler to implement and computationally more efficient. The RIPC scheme is not in anyway limited to implementation in a pseudo-spectral code or periodic boundary conditions, and can be used in complex geometries and with other suitable boundary conditions.
The purpose of this note is to correct some errors in the proofs of pressure error estimates in a... more The purpose of this note is to correct some errors in the proofs of pressure error estimates in and . The origin of the errors is the following incorrect inequality 1 (1) ), which the author incorrectly derived by identifying u V with u -1 for u ā H. 2 The correct inequality is: (2) However, the incorrect proofs induced by the error (1) can all be fixed as indicated below. More precisely, the proofs for the velocity error estimates in [1, 2] are all valid provided some minor changes of notations; the pressure error estimates still hold, but their proofs necessitate some additional estimates. In summary, all the results presented in remain valid provided some additional regularity assumptions on the exact solution are made. In the following we provide the details for the aforementioned modifications and corrections. This work was supported in part by NSF grant DMS-9205300 1 The incorrectness of (1) was pointed out to me by R. Temam to whom I am grateful. Later on J.L. Guermond has also expressed to me his doubts about (1) 2 The confusion can be partly explained by a technical sublety. We know that V ā H 1 0 (ā¦) d and hence H 1 0 (ā¦) d = V ā V ā„ . By the Riesz theorem and elementary properties of Hilbert spaces, V is isomorphic to and could be identified with a subspace of H -1 (ā¦) d . However this identification must not be made because, as it is usual with evolution equations, we already identified H with a subspace of V (V ā H ā V ). Hence this double identification is not allowed
We developed a numerical method for computing the effective properties of a microstructure. The m... more We developed a numerical method for computing the effective properties of a microstructure. The method is particularly efficient and accurate for microstructures with a diffuse-interface description similar to those generated from phase-field simulations. In particular, we considered the diffusive transport property of a microstructure by solving the steady-state diffusion equation using a Fourier-Chebyshev spectral method. Computed effective diffusivities agree very well with existing analytical solutions and computer simulations for a number of simple model systems. Combining with the phase-field model for simulating microstructure evolution, the proposed method can be applied for modeling the temporal evolution of effective properties. This is illustrated by considering grain growth and the corresponding effective transport property evolution as function of time.
A numerical simulation of the unsteady incompressible flow in the unit cavity is performed by usi... more A numerical simulation of the unsteady incompressible flow in the unit cavity is performed by using a Chebyshev-Tau approximation for the space variables. The high accuracy of the spectral methods and the condensed distribution of the Chebyshev-collocation points near the boundary enable us to obtain reliable results for high Reynolds numbers with a moderate number of modes. It is found that the flow converges to a stationary state for Reynolds numbers (Re) up to 10,000; for Reynolds numbers larger than a critical value 10,000 < Re 1 < 10,500 and less than another critical value 15,000 < Re 2 < 15,500, the flow becomes periodic in time which indicates a Hoph bifurcation; the flow loses time periodicity for Re > Re 2.
Two new Gauge-Uzawa schemes are constructed for incompressible flows with variable density. One i... more Two new Gauge-Uzawa schemes are constructed for incompressible flows with variable density. One is in the conserved form while the other is in the convective form. It is shown that the first-order versions of both schemes, in their semi-discretized form, are unconditionally stable. Numerical experiments indicate that the first-order (resp. second-order) versions of the two schemes lead to first-order (resp. second-order) convergence rate for all variables and that these schemes are suitable for handling problems with large density ratios such as in the situation of air bubble rising in water.
International Journal for Numerical Methods in Fluids, 2006
In Gresho and Sani (Int. J. Numer. Methods Fluids 1987; 7:1111ā1145; Incompressible Flow and the ... more In Gresho and Sani (Int. J. Numer. Methods Fluids 1987; 7:1111ā1145; Incompressible Flow and the Finite Element Method, vol. 2. Wiley: New York, 2000) was proposed an important hypothesis regarding the pressure Poisson equation (PPE) for incompressible flow: Stated there but not proven was a soācalled equivalence theorem (assertion) that stated/asserted that if the NavierāStokes momentum equation is solved simultaneously with the PPE whose boundary condition (BC) is the Neumann condition obtained by applying the normal component of the momentum equation on the boundary on which the normal component of velocity is specified as a Dirichlet BC, the solution (u, p) would be exactly the same as if the āprimitiveā equations, in which the PPE plus Neumann BC is replaced by the usual divergenceāfree constraint (ā Ā· u = 0), were solved instead.This issue is explored in sufficient detail in this paper so as to actually prove the theorem for at least some situations. Additionally, like the ori...
A simple and eective numerical method is proposed for simulating the temporal diusive mass transp... more A simple and eective numerical method is proposed for simulating the temporal diusive mass transport process through a microstructure with arbitrary complexity described by a phase-Ā®eld approach. The mass diusion through a given microstructure is modeled by a diusion equation with a variable diusion coecient, which is solved by an ecient and accurate semi-implicit spectral method. It is shown that it is possible to extract the eective diusion coecient for any given microstructure from the temporal concentration proĀ®les. The method is used to simulate the grain boundary diusion in a single-phase polycrystalline grain structure and the heterogeneous diusion in a twophase microstructure with dierent diusion coecient in each phase. Results are compared with existing analytical theories and computer simulations.
Journal of Computational and Applied Mathematics, 2014
This paper is concerned with approximation properties of orthonormal mapped Chebyshev functions (... more This paper is concerned with approximation properties of orthonormal mapped Chebyshev functions (OMCFs) in unbounded domains. Unlike the usual mapped Chebyshev functions which are associated with weighted Sobolev spaces, the OMCFs are associated with the usual (non-weighted) Sobolev spaces. This leads to particularly simple stiffness and mass matrices for higher-dimensional problems. The approximation results for both the usual tensor product space and hyperbolic cross space are established, with the latter particularly suitable for higher-dimensional problems.
The incompressible Stokes equations with prescribed normal stress (open) boundary conditions on p... more The incompressible Stokes equations with prescribed normal stress (open) boundary conditions on part of the boundary are considered. It is shown that the standard pressure-correction method is not suitable for approximating the Stokes equations with open boundary conditions, whereas the rotational pressure-correction method yields reasonably good error estimates. These results appear to be the first ever published for splitting schemes with open boundary conditions. Numerical results in agreement with the error estimates are presented.
A fast Chebyshev-Fourier algorithm for Poisson-type equations in polar geometries is presented in... more A fast Chebyshev-Fourier algorithm for Poisson-type equations in polar geometries is presented in this paper. The new algorithm improves upon the algorithm of Jie Shen (1997), by taking advantage of the odd-even parity of the Fourier expansion in the azimuthal direction, and it is shown to be more efficient in terms of CPU and memory.
Boundary perturbation methods, in which the deviation of the problem geometry from a simple one i... more Boundary perturbation methods, in which the deviation of the problem geometry from a simple one is taken as the small quantity, have received considerable attention in recent years due to an enhanced understanding of their convergence properties. One approach to deriving numerical methods based upon these ideas leads to Bruno and Reitich's generalization [Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), pp. 317-340] of Rayleigh and Rice's classical algorithm giving the "method of variation of boundaries" which is very fast and accurate within its domain of applicability. Treating problems outside this domain (e.g., boundary perturbations which are large and/or rough) led Nicholls and Reitich [Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), pp. 1411-1433] to design the "transformed field expansions" (TFE) method, and the rigorous numerical analysis of these recursions is the subject of the current work. This analysis is based upon analyticity estimates for the TFE expansions coupled to the convergence of Fourier-Legendre Galerkin methods. This powerful and flexible analysis is extended to a wide range of problems including those governed by Laplace and Helmholtz equations, and the equations of traveling free-surface ideal fluid flow.
We propose a new weighted weak formulation for the Fokker-Planck equation of the finitely extensi... more We propose a new weighted weak formulation for the Fokker-Planck equation of the finitely extensible nonlinear elastic dumbbell model and prove its well-posedness in weighted Sobolev spaces. We also propose simple and efficient semi-implicit time-discretization schemes which are unconditionally stable, i.e., the step size of time marching does not depend on the number of the bases used in configurational space. We then restrict ourselves to the two-dimensional case and construct two Fourier-Jacobi spectral-Galerkin algorithms which enjoy the following properties: (i) they are unconditionally stable, spectrally accurate, and of optimal computational complexity; (ii) they conserve the volume and provide accurate approximation to higher-order moments of the distribution function; and (iii) they can be easily extended to coupled nonhomogeneous systems. Numerical results are presented to show how to choose a proper weight to get the best numerical results of the distribution function and the polymer stress.
Mathematical modeling and numerical simulation of smectic C liquid crystals which possess the spo... more Mathematical modeling and numerical simulation of smectic C liquid crystals which possess the spontaneous polarization are considered in this paper. In particular, the model allows for a system with a zero net polarization which is one of the ubiquitous systems of the polarized liquid crystals. Theoretical and numerical investigations are conducted to study effects of the energy associated with the polarization, switching patterns between two uniform states by an externally applied field and random noise, as well as a relation between polarization and applied field near the phase transition from the smectic A and smectic C.
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