Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis (AMNA)

People Research Publications Teaching

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1. Propose far field boundary conditions for linear hyperbolic systems and a general tool of scaling the included factors
2. Well-posedness and regularity of the resulting initial boundary value problem
3. Establish a general criteria for the convergence in time to steady state
4. Stable and efficient numerical scheme to solve the resulting initial boundary value problem
5. Numerical examples supporting our choice of the factors

Summary

Several physical phenomena of great importance for applications are described by equations whose solutions are composed of waves. An important part of these problems are posed on unbounded domains. To compute a numerical solution to such problems, it is necessary, due to finite computational resources, to truncate the unbounded domain. This is done by introducing an artificial boundary Γ, defining a new domain Ω, which we will refer to as the computational domain. For the problem to be well-posed, it must be closed with a suitable boundary condition on Γ. Also, special care have to be taken when choosing the boundary condition, so that the solution on Ω will be close to the solution on the unbounded domain.

Often the artificial boundary Γ is placed in the far field where the solution is composed of linear waves traveling out of Ω. The fundamental observation is therefore that all reflections caused by the boundary condition on Γ will contaminate the solution in the interior. Hence, for linear waves, the boundary condition should absorb the energy at the artificial boundary.

The development of better boundary conditions is important, since it will allow for more accurate simulation of wave phenomena in many areas. One such area is aeroacoustics, which is the enabling science for control of acoustics in early design stages of aircraft, cars, and trains. Design for noise reduction in an aircraft is a typical example of a problem posed on an unbounded domain. Simulation of elastic waves, to predict strong ground motions, earthquakes and other geological phenomena, is another example.

The level of difficulty of constructing a particular boundary condition is determined by the underlying problem. In this project, we mainly consider boundary conditions for linear and nonlinear hyperbolic partial differential equation on an infinite domain. Ideally, these boundary conditions should prevent any nonphysical reflection of outgoing waves and should be easy to implement numerically. They should also present together with the governing equation a well-posed truncated problem which is a basic requirement for the corresponding numerical approximation to be stable. Example of hyperbolic equations include the Euler equations of gas dynamics, the shallow water equations, Maxwell's equations, and equations of magnetohydrodynamics. For these hyperbolic problems the correct boundary condition is that waves traveling across the boundary should not be reflected back. Boundary conditions with this property are often referred to as non-reflecting, transparent, artificial or absorbing boundary conditions (ABCs).

The theoretical basis for ABCs stems from a paper by Engquist and Majda which discusses both ideal ABCs and a method for constructing approximate forms. In addition, a paper by Kreiss which analyzes the well-posedness of the initial boundary value problems for hyperbolic systems. Many researchers have been active in this area in the last years. However, their work has been mainly concerned with ABCs that are better suited for a transient solution than for a steady solution, and most of these boundary conditions lead to steady solutions of poor accuracy.

In this project we are concerned with ABCs that lead to accurate steady solutions. In this context, Bayliss and Turkel derived non-reflecting conditions for the Euler equations which they used for steady state calculations. These boundary conditions are obtained from expansions of the solution at large distances. Accurate boundary conditions for the steady Euler equations in a channel were also studied by Giles.

Engquist and Halpern constructed a new class of boundary conditions that combine the properties of ABCs for transient solutions and the properties of transparent boundary conditions for steady state problems. These boundary conditions, which called far field boundary conditions (FBCs), can be used in both the transient regime and when the solution approaches the steady state. In this sense, they can be applied when the evanescent and traveling waves are present in the time-dependent calculation or when a time-dependent formulation is used for computations to steady state. These combined technique is used for the Euler equations in the periodic channel by Giles. In case of hyperbolic systems, these FBCs are defined up to matrix factor in front of the steady terms. This poses the following computational problem (which is one of the main subjects of this project): How to choose this factor in a way to accelerate the convergence to the steady state, and to improve the accuracy of the transient solution.

We observe that the FBCs simulate the radiation of energy out of Ω. An incorrect specification of these boundary conditions can cause spurious reflected waves to be generated at Γ. These waves represent energy propagating into Ω. Since they are not part of the desired solution, they can substantially reduce the accuracy of the computed solution. On the other hand, if the time-dependent equations are only an intermediate step toward computing a steady state, then a flow of energy into Ω can delay the convergence to the steady state. Conversely, the correct specification of FBCs can accelerate the convergence. Thus, an answer of the above question consists in minimizing the spurious reflections.

German team:

• Matthias Ehrhardt

Palestinian team:

• Tareq Amro

External Cooperation with Austrian team:

• Anton Arnold

German institutions:

• Department of Mathematics and Informatics, Faculty of Mathematics and Natural Sciences, University of Wuppertal.

Palestinian institutions:

• Faculty of Science and Technology, Hebron University,

Publications of the Project

[AAE]	T. Amro, A. Arnold and M. Ehrhardt,
	Optimized Far Field Boundary Conditions for Hyperbolic Systems, Preprint No 12/02, University of Wuppertal, January 2012.

References

1. T. Amro, Far Field Boundary Conditions and Perfectly Matched Layer for some Wave Propagation Problems, Dissertation, Westfälische Wilhelms-Universität Münster, 2007.
2. X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations, Commun. Comput. Phys. 4 (2008), 729-796. (open-access article)
3. D. Appelö, T. Hagstrom and G. Kreiss, Perfectly matched layer for hyperbolic systems: general formulation, well-posedness and stability, SIAM Appl. Math. 67 (2006), 1-23.
4. A. Bayliss and E. Turkel, Far field boundary conditions for compressible flows, J. Comput. Phys. 48 (1982), 182-199.
5. M. Ehrhardt, Absorbing Boundary Conditions for Hyperbolic Systems, Numer. Math. Theor. Meth. Appl. 3 (2010), 295-337.
6. B. Engquist and L. Halpern, Far field boundary conditions for computation over long time, Appl. Numer. Math. 4 (1988), 21-45.
7. T. Hagstrom, Radiation Boundary Condition Page
8. T. Hagstrom and T. Warburton, Complete Radiation Boundary Conditions: Minimizing the Long Time Error Growth of Local Methods, SIAM J. Numer. Anal. 47 (2009), 3678-3704.
9. T. Hagstrom, D. Givoli and T. Warburton, Radiation boundary conditions for time-dependent waves based on complete plane wave expansions, J. Comput. Appl. Math. 234 (2010), 1988-1995.