A space-time BIE method for nonhomogeneous exterior wave equation problems. The Dirichlet case

Falletta, Silvia; Monegato, Giovanni; Scuderi, Letizia

doi:10.1093/imanum/drr008

Cited by 33 publications

(84 citation statements)

References 21 publications

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“…We start by briefly recalling the main steps of the Lubich-collocation method for the discretization of the NRBC (2) (for more details we refer to [10] and [3]). …”

Section: Discretization Of the Nrbcmentioning

confidence: 99%

“…The time integrals appearing in the definition of the single and double layer operators are discretized by means of the convolution quadrature formula associated with the second order Backward Differentiation Method (BDF) for ordinary differential equations (see [3]):…”

Section: Discretization Of the Nrbcmentioning

confidence: 99%

“…In the first case, to test the accuracy of the approximation obtained by the fictitious domain approach, we construct a reference "exact" solution by applying the Lubich-collocation boundary element method described in [3] with a very fine space and time discretization. Once the density function is retrieved, the solution at any point in the infinite domain is obtained by computing the associated potential (see [3] for details). This solution will be denoted by the acronym BEM.…”

Section: Finite Element Approximation and Time Discretization Of The mentioning

confidence: 99%

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A Fictitious Domain Approach for Wave Propagation Problems in Unbounded Domains

Falletta¹,

Monegato²

2015

Proceedings of the 5th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. We consider the scattering of waves by an obstacle O ⊂ R 2 having a sufficiently smooth boundary Γ. By using a fictitious domain approach, we artificially extend the solution inside the obstacle O. Then we solve the original problem in the simpler domain that includes O and that is artificially bounded by a perfectly transparent boundary B, delimiting the region of interest. We prescribe on B a Non Reflecting Boundary Condition (NRBC) which is based on a space-time integral equation and that defines a relationship that the solution of the differential problem and its normal derivative must satisfy on B (see [2]). Further, we enforce the Dirichlet boundary condition on Γ in a weak sense, by means of Lagrange multipliers. The NRBC is discretized on B by combining a special second order (in time) convolution quadrature and a standard collocation method in space. In the enlarged domain, this discretization is coupled with an unconditionally stable ODE time integrator and a FEM in space. The constraint on Γ is imposed by a matrix B h that represents a discrete trace operator.A particularly useful application of this approach is the scattering of a wave by rotating rigid bodies. In this case the method avoids the complexity of constructing at each time step a new finite element computational mesh and requires only the construction of the discrete trace operator B h . We present some results we have obtained for rotating (even multiple) scatterers and non trivial data. 959

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“…We start by briefly recalling the main steps of the Lubich-collocation method for the discretization of the NRBC (2) (for more details we refer to [10] and [3]). …”

Section: Discretization Of the Nrbcmentioning

confidence: 99%

Section: Discretization Of the Nrbcmentioning

confidence: 99%

Section: Finite Element Approximation and Time Discretization Of The mentioning

confidence: 99%

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A Fictitious Domain Approach for Wave Propagation Problems in Unbounded Domains

Falletta¹,

Monegato²

2015

Proceedings of the 5th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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“…Among the most popular methods for discretizing this equation are: a) the convolution quadrature (CQ) method [29], [30], [22], [28], [5], [12] and b) the direct space-time Galerkin discretization of (2) (see, e.g., [4], [18], [19], [35], [34], [38]). …”

Section: Introductionmentioning

confidence: 99%

The panel-clustering method for the wave equation in two spatial dimensions

Falletta

Sauter

2016

Journal of Computational Physics

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We consider the numerical solution of the wave equation in a two-dimensional domain and start from a boundary integral formulation for its discretization. We employ the convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. Our main focus is the sparse approximation of the arising sequence of boundary integral operators by panel clustering. This requires the definition of an appropriate admissibility condition such that the arising kernel functions can be efficiently approximated on admissible blocks.The resulting method has log-linear complexity O N (N + M ) q 4+s , s ∈ {0, 1}, where N is the number of time points, M denotes the dimension of the boundary element space, and q = O (log N + log M ) is the order of the panelclustering expansion.Numerical experiments will illustrate the efficiency and accuracy of the proposed CQ-BEM method with panel clustering.

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“…The discretization then consists of approximating the (time-depending) differential equation in the Laplace domain by a time stepping method -besides multisteps methods also Runge-Kutta methods have been proposed and analyzed for this purpose [12,13,15,3,1,2,5]. The transformation back to the time domain results in a discrete convolution equation which then can be solved numerically.…”

Section: Introductionmentioning

confidence: 99%

Generalized convolution quadrature based on Runge-Kutta methods

López-Fernández

Sauter

2015

Numer. Math.

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Convolution equations for time and space-time problems have many important applications, e.g., for the modelling of wave or heat propagation via ordinary and partial differential equations as well as for the corresponding integral equation formulations.For their discretization, the convolution quadrature (CQ) has been developed since the late 1980's and is now one of the most popular method in this field.However, the method and the theory are restricted to constant time stepping and only recently the implicit Euler -generalized convolution quadrature (gCQ) has been developed which allows for variable time stepping.In this paper, we develop the gCQ for Runge-Kutta methods with variable time stepping and present the corresponding stability and convergence analysis. For this purpose, some new theoretical tools such as tensorial divided differences, summation by parts with Runge-Kutta differences and a calculus for Runge-Kutta discretizations of generalized convolution operators such as an associativity property will be developed in this paper.Numerical examples will illustrate the stable and efficient behavior of the resulting discretization.

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A space-time BIE method for nonhomogeneous exterior wave equation problems. The Dirichlet case

Cited by 33 publications

References 21 publications

A Fictitious Domain Approach for Wave Propagation Problems in Unbounded Domains

A Fictitious Domain Approach for Wave Propagation Problems in Unbounded Domains

The panel-clustering method for the wave equation in two spatial dimensions

Generalized convolution quadrature based on Runge-Kutta methods

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