| 1. |
- Engquist, Björn, et al.
(författare)
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Projection shock capturing algorithms
- 1990
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Ingår i: Twelfth International Conference on Numerical Methods in Fluid Dynamics. - Berlin : Springer-Verlag. ; , s. 335-336
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Konferensbidrag (refereegranskat)
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| 2. |
- Andersson, U, et al.
(författare)
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A contribution to wavelet-based subgrid modeling
- 1999
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Ingår i: Applied and Computational Harmonic Analysis. - 1063-5203 .- 1096-603X. ; 7:2, s. 151-164
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Tidskriftsartikel (refereegranskat)abstract
- A systematic technique for the derivation of subgrid scale models in the numerical solution of partial differential equations is described. The technique is based on Haar wavelet projections of the discrete operator followed by a sparse approximation. As numerical testing suggests, the resulting numerical method will accurately represent subgrid scale phenomena on a coarse grid. Applications to numerical homogenization and wave propagation in materials with subgrid inhomogeneities are presented.
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| 3. |
- Dorobantu, M, et al.
(författare)
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Wavelet-based numerical homogenization
- 1998
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Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 35:2, s. 540-559
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Tidskriftsartikel (refereegranskat)abstract
- A numerical homogenization procedure for elliptic differential equations is presented. It is based on wavelet decompositions of discrete operators in find and coarse scale components followed by the elemination of the fine scale contributions. If the operator is in divergence form, this is preserved by the homogenization procedure. For periodic problems, results similar to classical effective coefficient theory is proved. The procedure can be applied to problems that are not cell-periodic.
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| 4. |
- Durlofsky, L. J., et al.
(författare)
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Triangle based adaptive stencils for the solution of hyperbolic conservation laws
- 1992
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Ingår i: Journal of Computational Physics. - : Academic Press. - 0021-9991 .- 1090-2716. ; 98:1, s. 64-73
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Tidskriftsartikel (refereegranskat)abstract
- A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.
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| 5. |
- Eliasson, P, et al.
(författare)
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The effect of dissipation and coarse grid resolution for multigrid in flow problem
- 1996
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Konferensbidrag (refereegranskat)abstract
- This paper is to investigate the effects of the numerical dissipation and the resolution of the solution on coarser grids for multigrid with the Euler equation approximations. The convergence is accomplished by multi-stage explicit time-stepping to steady state accelerated by FAS multigrid. A theoretical investigation is carried out for linear hyperbolic equations in one and two dimensions. The spectra reveals that for stability and hence robustness of spatial discretizations with a small amount of numerical dissipation the grid transfer operators have to be accurate enough and the smoother of low temporal accuracy. Numerical results give grid independent convergence in one dimension. For twodimensional problems with a small amount of numerical dissipation, however, only a few grid levels contribute to an increased speed of convergence. This is explained by the small numerical dissipation leading to dispersion. Increasing the mesh density and hence making the problem over resolved increases the number of mesh levels contributing to an increased speed of convergence. If the steady state equations are elliptic, all grid levels contribute to the convergence regardless of the mesh density
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| 6. |
- Engquist, Björn, et al.
(författare)
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Absorbing boundary conditions for domain decomposition
- 1998
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Ingår i: Applied Numerical Mathematics. - 0168-9274 .- 1873-5460. ; 27:4, s. 315-324
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we would like to point out some similarities between two artificial boundary conditions. One is the far field or absorbing boundary conditions for computations over unbounded domain. The other is the boundary conditions used at the boundary between subdomains in domain decomposition. We show some convergence result for the generalized Schwarz alternating method (GSAM), in which a convex combination of Dirichlet data and Neumann data is exchanged at the artificial boundary. We can see clearly how the mixed boundary condition and the relative size of overlap will affect the convergence rate. These results can be extended to more general coercive elliptic partial differential equations using the equivalence of elliptic operators. Numerically first- and second-order approximations of the Dirichlet-to-Neumann operator are constructed using local operators, where information tangential to the boundary is included. Some other possible extensions and applications are pointed out. Finally numerical results are presented.
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| 7. |
- Engquist, Björn, et al.
(författare)
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Absorbing boundary conditions for the numerical simulation of waves
- 1997
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Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 1, s. 629-651
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Tidskriftsartikel (refereegranskat)abstract
- In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation. Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artifical boundaries. These boundary conditions not only guarantee stable difference approximations, but also minimize the (unphysical) artificial reflections that occur at the boundaries.
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| 8. |
- Engquist, Björn, et al.
(författare)
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Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients
- 1997
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Ingår i: SIAM Journal on Numerical Analysis. - : SIAM Publications. - 0036-1429 .- 1095-7170. ; 34:6, s. 2254-2273
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Tidskriftsartikel (refereegranskat)abstract
- Standard multigrid methods are not so effective for equations with highly oscillatory coefficients. New coarse grid operators based on homogenized operators are introduced to restore the fast convergence rate of multigrid methods. Finite difference approximations are used for the discretization of the equations. Convergence analysis is based on the homogenization theory. Proofs are given for a two-level multigrid method with the homogenized coarse grid operator for two classes of two-dimensional elliptic equations with Dirichlet boundary conditions.
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| 9. |
- Engquist, Björn, et al.
(författare)
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Erding New coarse grid operators for highly oscillatory coefficient elliptic problems
- 1996
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Ingår i: Journal of Computational Physics. - 0021-9991 .- 1090-2716. ; 2, s. 296-306
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Tidskriftsartikel (refereegranskat)abstract
- New coarse grid operators are developed for elliptic problems with highly oscillatory coefficients. The new coarse grid operators are constructed directly based on the homogenized differential operators or hierarchically computed from the finest grid. A detailed description of this construction is provided. Numerical calculations for a two dimensional elliptic model problem show that the homogenized form of the equations is very useful in the design of coarse grid operators for the multigrid method. A more realistic problem of heat conduction in a composite structure is also considered.
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| 10. |
- Engquist, Björn, et al.
(författare)
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Fast wavelet based algorithms for linear evolution equations
- 1994
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 15:4, s. 755-775
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Tidskriftsartikel (refereegranskat)abstract
- The authors devise a class of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin [Comm. Pure Appl. Math., 44 (1991), pp. 141-1841, which they applied to general Calderon-Zygmund type integral operators. The authors apply a modification of their idea to linear hyperbolic and parabolic equations, with spatially varying coefficients. The complexity for hyperbolic equations in one dimension is reduced from O(N2) to O(N log3 N). There are somewhat better gains for parabolic equations in multidimensions
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| 11. |
- Engquist, Björn, et al.
(författare)
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High Order shock capturing methods
- 1995
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Ingår i: Computational Fluid Dynamics Review. - New York : John Wiley & Sons. - 0471955892 ; , s. 210-233
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Bokkapitel (refereegranskat)
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| 12. |
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| 13. |
- Engquist, Björn, et al.
(författare)
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Iterative gradient-Newton type methods for steady shock computations
- 1991
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Ingår i: SIAM. ; , s. 60-75
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Tidskriftsartikel (refereegranskat)abstract
- A class of modified Newton´s methods are applied to difference approximations of the two-dimensional steady Burgers´ equation and the transonic small disturbance equation. The solutions have sharp gradients which corresponds to boundare layers and shock waves in fluid dynamics. The nonlinear terms in the differential equations are approximated by modern shock capturing schemes. The regularity of the coefficients is analyzed theoretically and its effect on the convengence on the Newton´s method is studied numerically. Computational results from different types of gradient iterative methods and different types of preconditioners are presented. These methods are applied to the linear systems of the Newton iteration. The relative residuals in the Newton iterations are controlled such that a superlinear rate of convergence is preserved
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| 14. |
- Engquist, Björn, et al.
(författare)
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Large time behavior and homogenization of solutions of two-dimensional conservation laws
- 1993
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 46:1, s. 1-26
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Tidskriftsartikel (refereegranskat)abstract
- We study the large time behavior of solutions of scalar conservation laws in one and two space dimensions with periodic initial data. Under a very weak nonlinearity condition, we prove that the solutions converge to constants as time goes to infinity. Even in one space dimension our results improve the earlier ones since we only require the fluxes to be nonlinear in a neighborhood of the mean value of the initial data. We then use these results to study the homogenization problem for scalar conservation laws with oscillatory initial data.
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| 15. |
- Engquist, Björn, et al.
(författare)
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Long-time behavior of absorbing boundary conditions
- 1990
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Ingår i: Mathematical methods in the applied sciences. - : John Wiley & Sons. - 0170-4214 .- 1099-1476. ; 13:3, s. 189-203
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Tidskriftsartikel (refereegranskat)abstract
- A new class of computational far-field boundary conditions for hyperbolic partial differential equations was recently introduced by the authors. These boundary conditions combine properties of absorbing conditions for transient solutions and properties of far-field conditions for steady states. This paper analyses the properties of the wave equation coupled with these new boundary conditions: well-posedness, dissipativity and convergence in time.
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| 16. |
- Engquist, Björn, et al.
(författare)
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Multi-phase computations in geometrical optics
- 1996
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427 .- 1879-1778. ; 74:1-2, s. 175-192
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Tidskriftsartikel (refereegranskat)abstract
- In this work we propose a new set of partial differential equations (PDEs) which can be seen as a generalization of the classical eikonal and transport equations, to allow for solutions with multiple phases. The traditional geometrical optics pair of equations suffer from the fact that the class of physically relevant solutions is limited. In particular, it does not include solutions with multiple phases, corresponding to crossing waves. Our objective has been to generalize these equations to accommodate solutions containing more than one phase. The new equations are based on the same high frequency approximation of the scalar wave equation as the eikonal and the transport equations. However, they also incorporate a finite superposition principle. The maximum allowed number of intersecting waves in the solution can be chosen arbitrarily, but a higher number means that a larger system of PDEs must be solved. The PDEs form a hyperbolic system of conservation laws with source terms. Although the equations are only weakly hyperbolic, and thus not well-posed in the strong sense, several examples show the viability of solving the equations numerically. The technique we use to capture multivalued solutions is based on a closure assumption for a system of equations representing the moments.
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| 17. |
- Engquist, Björn, et al.
(författare)
-
Multigrid methods for differential equations with highly oscillatory coefficients
- 1993
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Konferensbidrag (refereegranskat)abstract
- New coarse grid multigrid operators for problems with highly oscillatory coefficients aredeveloped. These types of operators are necessary when the characters of the differentialequations on coarser grids or longer wavelengths are different from that on the fine grid.Elliptic problems for composite materials and different classes of hyperbolic problems arepractical examples.The new coarse grid operators can be constructed directly based on the homogenizeddifferential operators or hierarchally computed from the finest grid. Convergence analysisbased on the homogenization theory is given for elliptic problems with periodic coefficientsand some hyperbolic problems. These are classes of equations for which there exists afairly complete theory for the interaction between shorter and longer wavelengths in theproblems. Numerical examples are presented.
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| 18. |
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| 19. |
- Engquist, Björn, et al.
(författare)
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Numerical methods for oscillatory solutions to hyperbolic problems
- 1993
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Ingår i: Communications on Pure and Applied Mathematics. - : John Wiley & Sons. - 0010-3640 .- 1097-0312. ; 46:10, s. 1327-1361
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Tidskriftsartikel (refereegranskat)abstract
- Difference approximations of hyperbolic partial differential equations with highly oscillatory coefficients and initial values are studied. Analysis of strong and weak convergence is carried out in the practically interesting case when the discretization step sizes are essentially independent of the oscillatory wave lengths
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| 20. |
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| 21. |
- Engquist, Björn, et al.
(författare)
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The convergence rate of finite difference schemes in the presence of shocks
- 1998
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Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 35:6, s. 2464-2485
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Tidskriftsartikel (refereegranskat)abstract
- : Finite difference approximations generically have O(1) pointwise errors close to a shock. We show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher-order methods. Analytic and numerical examples of this form of accuracy are given. We also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock.
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| 22. |
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| 23. |
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| 24. |
- Engquist, Björn
(författare)
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Wavelet based numerical homogenization
- 1998
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Konferensbidrag (refereegranskat)abstract
- Classical homogenization is an analytic technique for approximating multiscale differential equations. The numbers of scales are reduced and the resulting equations are easier to analyze or numerically approximate. The class of problems that classical homogenization applies to is quite restricted. We shall describe a numerical procedure for homogenization, which starts from a discretization of the multiscale differential equation. In this procedure the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The wavelet homogenization applies to a wider class of problems than classical homogenization. The projection procedure is general and we give a presentation of a framework in Hilbert space, which also applies to the differential equation directly. The wavelet based homogenization technique is applied to discretizations of the Helmholtz equation. In one problem from electromagnetic compatibility a subgrid scale geometrical detail is represented on a coarser grid. In another a wave-guide filter is efficiently approximated in a lower dimension. The technique is also applied to the derivation of effective equations for a nonlinear problem and to the derivation of coarse grid operators in multigrid. These multigrid methods work very well for equations with highly oscillatory or discontinuous coefficients.
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| 25. |
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| 26. |
- Fatemi, E, et al.
(författare)
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Numerical solution of the high frequency asymptotic expansion for the scalar wave equation
- 1995
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Ingår i: Journal of Computational Physics. - : Academic Press. - 0021-9991 .- 1090-2716. ; 120:1, s. 145-155
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Tidskriftsartikel (refereegranskat)abstract
- New numerical methods are derived for calculation of high frequency asymptotic expansion of the scalar wave equation. The nonlinear partial differential equations defining the terms in the expansion are approximated directly rather than via ray tracing, High resolution numerical algorithms are used to handle discontinuities and new devices are introduced to represent the multivalued character of the solution.
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| 27. |
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| 28. |
- Harten, A, et al.
(författare)
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Uniformly High Order Accurate Essentially Non-oscillatory Schemes, III
- 1997
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 131:1, s. 3-47
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Tidskriftsartikel (refereegranskat)abstract
- We continue the construction and the analysis of essentially non-oscillatory shock capturing methods for the approximation of hyperbolic conservation laws. We present an hierarchy of uniformly high-order accurate schemes which generalizes Godunov's scheme and its second-order accurate MUSCL extension to an arbitrary order of accuracy. The design involves an essentially non-oscillatory piecewise polynomial reconstruction of the solution from its cell averages, time evolution through an approximate solution of the resulting initial value problem, and averaging of this approximate solution over each cell. The reconstruction algorithm is derived from a new interpolation technique that, when applied to piecewise smooth data, gives high-order accuracy whenever the function is smooth but avoids a Gibbs phenomenon at discontinuities. Unlike standard finite difference methods this procedure uses an adaptive stencil of grid points and, consequently, the resulting schemes are highly nonlinear.
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| 29. |
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| 30. |
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| 31. |
- Ton, V, et al.
(författare)
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Numerical simulation of high speed chemically reacting flow
- 1994
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Ingår i: Theoretical and Computational Fluid Dynamics. - 0935-4964 .- 1432-2250. ; 6:2-3, s. 161-179
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Tidskriftsartikel (refereegranskat)abstract
- The essentially nonoscillatory (ENO) shock-capturing scheme for the solution of hyperbolic equations is extended to solve a system of coupled conservation equations governing two-dimensional, time-dependent, compressible chemically reacing flow with full chemistry. The thermodynamic properties of the mixture are modeled accurately, and stiff kinetic terms are separated from the fluid motion by a fractional step algorithm. The methodology is used to study the concept of shock-induced mixing and combustion, a process by which the interaction of a shock wave with a jet of low-density hydrogen fuel enhances mixing through streamwise vorticity generation. Test cases with and without chemical reaction are explored here. Our results indicate that, in the temperature range examined, vorticity generation as well as the distribution of atomic species do not change significantly with the introduction of a chemical reaction and subsequent heat release. The actual diffusion of hydrogen is also relatively unaffected by the reaction process. This suggests that the fluid mechanics of this problem may be successfully decoupled from the combustion processes, and that computation of the mixing problem (without combustion chemistry) can elucidate much of the important physical features of the flow.
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| 32. |
- Weinan, E, et al.
(författare)
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Blowup of solutions of the unsteady Prandtl's equation
- 1997
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Ingår i: Communications on Pure and Applied Mathematics. - 0010-3640 .- 1097-0312. ; 50, s. 1287-1293
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Tidskriftsartikel (refereegranskat)abstract
- We prove that for certain class of compactly supported C˜ initial data, smooth solutions of the unsteady Prandtl's equation blow up in nite time
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