# Single exponential decay equation. Single exponential decay equation

The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature.

This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known.

Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.

### Inhaltsverzeichnis

As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations.

Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm.

The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems: the need to construct many dense matrices and to evaluate many singular and near-singular integrals.

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The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method HF FMM and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times.

The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested.

### Single exponential decay function

The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled.

### Graphing Exponential Functions

The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity.

Single exponential decay equation efficiency of the approach is confirmed by extensive numerical experiments. Freie Schlagwörter DE Wellengleichung, retardierte Potential, Randelementmethode, Zeitbereichs-Randintegralgleichung, Faltungsquadratur, Runge-Kutta-Verfahren, hierarchische Matrizen, Fast Multipole Methoden Freie Schlagwörter EN wave equation, retarder potential, boundary element method, time-domain boundary integral equation, convolution quadrature, Runge-Kutta method, hierarchical matrices, fast multipole methods, data-sparse techniques Klassifikation DDC.