B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31(139): 629--651, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....second order FD scheme needs to choose only 4 5 points per wavelength (in average) for a reasonable accuracy for 2D problems. 6. Absorbing Boundary Condition We suggest an absorbing boundary condition (ABC) which e ectively absorbs incidents waves. When K = v, the conventional rst order ABC [2, 3] in (1.1.b) often selects = v: But the resulting ABC introduces a larger amount of re ection as the angle of the incident wave (measured from the outer normal) grows. More e ective ABCs can be selected taking into account the incident angles. Since the incident angle of wavefronts is not known ....

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), pp. 629-651.

....efficient nonreflecting boundary conditions. Waves should impinge on the (artificial) outer boundary and disappear without reflection, but they do not. For long time integration, the reflected energy can exceed that from the true field. This is true of both local differential boundary conditions [5, 10] and (finite) absorbing regions [7, 27] While there is an extensive literature on the subject (see the review articles [14, 23] only a few methods have been proposed which are exact. Ting and Miksis suggested an interesting approach [41] that relies on Kirchhoff s formula, but which is ....

....the fast algorithm for handling the convolution operators that arise. In Section 4, we present simple temporal and spatial discretization schemes, and in Section 5, we present a number of numerical experiments. We compare the performance of our exact scheme, local Engquist Majda conditions [10], and the recently popular PML method [7] which uses an absorbing region to dampen undesired reflections. Our conclusions and directions for future work are discussed in Section 6. 2. EXACT NONREFLECTING BOUNDARY CONDITIONS Let us consider the wave equation #u (1) in the exterior domain ....

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B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput. 31, 629 (1977).

....leads to an approximate problem, which is defined in a bounded domain G. Here, we introduce a circular artificial boundary of radius R, which is assumed to be large enough, and use the first order and second order absorbing boundary conditions constructed in the case of polar coordinates [14]. For the macro hybrid formulation, the domain G is decomposed into two parts G 1 and G 2 with a circular interface Gamma j Gamma 12 (see Figure 3) The boundary value problem considered here is of the form (2.1) where c is now a negative constant, f j 0; and oe is a complex valued ....

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), pp. 629--651.

....will only be perfectly nonreflecting when the outgoing waves have wavecrests which are aligned with the boundary. To minimize the reflection when the wave incidence is not normal it would be possible to employ higher order methods which have been successfully developed for the Euler equations [20 22]. Finally, for cases in which the particular choice of boundary conditions is determined by other factors, an interesting possibility would be to incorporate boundary condition considerations into the design of the preconditioner, so that the combination of the preconditioner and the boundary ....

B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput. 31, 629 (1977).

.... in electromagnetics parallels the search for open boundary conditions in computational fluid mechanics [51, 50] Absorbing boundary conditions have been used for over twenty years in electromagnetic codes to absorb outgoing waves at ## and to ensure that reflected waves do not corrupt the solution [41]. As a first order approximation, absorbing boundary conditions correspond to Sommerfeld s radiation condition for the Helmholtz equation; at high frequencies where greater accuracy may be required, higher order absorbing boundary condition are also possible [58, 57, 69] A di#erent class of ....

....do not exist in nature, they are reasonable approximations and are used in many electromagnetic models for their simplicity [11, 63] A standard first order absorbing boundary condition is of the form ( n H) n g (x ##) 2. 24) where g is the tangential trace of some known field [41, 97]. The particular absorbing boundary condition (2.24) ensures that electromagnetic waves normally incident on the boundary will be completely absorbed. This is more physically reasonable than a PEC or PMC, although there will be some reflections due to waves striking the boundary obliquely. ....

B. Engquist and A. Madja. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31:1--24, 1977.

....terms of computer resources due to the non locality in time, the dispersion relation between time and space variables may be rational approximated in the dual domain. This construction scheme has been successfully applied by a number of authors. Following the pioneering work of Engquist and Majda [4] on hyperbolic equations, advanced approximation techniques have been proposed for mixed parabolic hyperbolic systems (Halpern, 6] or parabolic equations (Hagstrom, 5] Nonlocal boundary conditions. However, there are a number of problems acting very sensitive with respect to the magnitude of ....

B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Mathematics of Computation, 31(139):629--651, July 1977.

....of U PML, and the e#ect is still better than using a classical nonreflecting boundary condition to terminate the U PML layer. The Gems time domain 3D code includes PML, U PML for dispersive materials, and also the first order Mur ABC [74] The Mur scheme is the application of the Engquist Majda [31] ABC to the Maxwell equations. Other approaches to ABC are still being explored in the search for a cheaper ABC, for example [42, 82] We have not explored this area. One interesting approach is to use the plane wave time domain (PWTD) method [93] as ABC. This would make it possible to put the ....

....the ionized channel where the current is propagating. Another example is for wires running through a PEC surface without Galvanic contact and where the other side of the PEC surface is excluded from the simulation. Since the wire is discretized by a 1D leap frog scheme the one way wave equation [31] is perfectly suitable. We get 2 ) 8.35) N 1 N 1 ) 8.36) 8.6 Discrete distribution The wire equations and the Maxwell equations contains two interaction terms, Jw and E # , coupling the two system of equations in (8.19) together. For a wire that is discretized ....

B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comput., 31(139):629--651, 1977.

....U PML, and the e#ect is still better than using a classical nonreflecting boundary condition to terminate the U PML layer. The Gems time domain 3D code includes PML, U PML for dispersive materials, and also the first order Mur ABC [Mur81] The Mur scheme is the application of the Engquist Majda [EM77] ABC to the Maxwell equations. Other approaches to ABC are still being explored in the search for a cheaper ABC, for example [GK98, Ram98] We have not explored this area. One interesting approach is to use the plane wave time domain (PWTD) method (see Section 1.1.3) SEAM00] as ABC. This would ....

B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comput., 31(139):629--651, 1977.

....away, the boundaries are typically less than one chord away for most turbomachinery applications. This situation may lead to computational inaccuracies if the boundary conditions are not suitably formulated. Various techniques have been developed to minimize the reflection of the out going waves [19, 20, 21, 22] and an overview is given by Givoli [23] Here two different set of treatments, one for steady state computation and the other for unsteady computation, are used. The steady state boundary treatment is based on the characteristics of the Euler equations. In particular, the steady state boundary ....

Engquist, B. and Majda, A. Absorbing Boundary Conditions for The Nu- merical Simulation of Waves. Mathematics of Computation, 31:629 651, 1977.

....and sensitive to parameter variations, such as grid stretching. Moreover, it appears that the complexity of the computational problem, i.e. the performance of the iterative solver, depends on the type of lateral boundary conditions employed. The dispersion relations and the theory developed in [8] can serve in the development Recommendations for future research 31 of suitable highly absorbing lateral boundary conditions for the free surface flow problem. A further attempt should be made to derive such highly absorbing boundary conditions. 32 ....

B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31:629-651, 1977.

....(say, infinity) from the interior region. For interior schemes involving traveling waves, the absorbing boundary condition must have the ability to absorb waves incident on it rather than reflecting them back into the interior of the domain. 2.1.1. Engquist and Majda approach. In their paper [2], Engquist and Majda proposed a pseudodi#erential operator which acts as a perfectly absorbing boundary condition for the scalar wave equation # 2 u #t 2 # 2 u #x 2 # 2 u #y 2 = 0 (2.1) with the related dispersion relation # 2 = k 2 l 2 . 2.2) The exact absorbing ....

....has been done by Kreiss [17] Sakamoto [22] and others. The well posedness of absorbing boundary conditions for the wave equation in particular has been considered by Trefethen and Halpern [26] The well posedness properties of particular absorbing boundary conditions are considered in [1] [2], 10] 12] 14] 21] by their authors. The well posedness theory is closely related to the stability of finite di#erence approximations of initial boundary value problems for hyperbolic equations. The stability criterion for hyperbolic initial boundary value problems is outlined by Gustafsson, ....

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B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), pp. 629--651.

....at the flow velocity u to leave the computational domain with minimal reflection in order to mimic an infinite domain. Otherwise the reflections will corrupt the results. In the inviscid flow regions, the non reflecting boundary conditions developed for the Euler equations can be used [43], 44] 45] 46] 47] 48] 49] 50] pp. 387 395, 51] 52] 53] 54] 55] 14] For the Navier Stokes equations more information is required at the artificial boundaries than for the Euler equations [56] 57] 58] 59] 60] 61] 62] For low Mach number flow, the acoustic waves ....

....towards the outlet. Figure 8: Acoustic pressure for acoustic pulse reflected at the inlet computed with reflecting boundary conditions. The non reflecting boundary conditions require the incoming characteristic variables not to change and the outgoing ones to be determined from the interior [43]. With nonreflecting boundary conditions, the inlet becomes permeable for the acoustic wave. It traverses the inlet with almost no reflection, as Fig. 9 illustrates. The acoustic pressure after = 1:4 is almost zero. Actually, it is about 5 orders of magnitude lower than with reflecing boundary ....

B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Mathematics of Computation, 31:629--651, 1977.

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B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31(139): 629--651, 1977.

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B. Engquist and A. Majda, "Absorbing Boundary Conditions for the Numerical Simulation of Waves," Mathematics of Computation 31(139), pp. 629--651, 1977.

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B. Engquist, A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., Volume 31, number 139, p.629--651, 1977.

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B. Engquist, A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., Volume 31, number 139, p.629-651, 1977.

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B. Engquist, A. Majda. Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., Volume 31, number 139, p.629-651, (1977).

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B. Engquist, A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., Volume 31, number 139, p.629--651, 1977.

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B. Engquist, A. Majda. Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., Volume 31, number 139, p.629-651, (1977).

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B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves," Math. Computation, vol. 31, pp. 629--651, 1977.

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B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves," Math. Comput., vol. 31, pp. 629-651, July 1977.

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B. ENGQUIST AND A. MAJDA, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), pp. 629--651.

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B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves," Math. Computation, vol. 31, pp. 629--651, 1977.

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B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31#139#:629#651, 1977.

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Engquist, B. and A. Majda, 1977 : Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31, no 139, 629-651.

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