## Fachbereich Mathematik

- Geometric Ergodicity of Binary Autoregressive Models with Exogenous Variables (2013)
- In this paper we introduce a binary autoregressive model. In contrast to the typical autoregression framework, we allow the conditional distribution of the observed process to depend on past values of the time series and some exogenous variables. Such processes have potential applications in econometrics, medicine and environmental sciences. In this paper, we establish stationarity and geometric ergodicity of these processes under suitable conditions on the parameters of the model. Such properties are important for understanding the stability properties of the model as well as for deriving the asymptotic behavior of the parameter estimators.

- A Nonlinear Galerkin Scheme Involving Vector and Tensor Spherical Harmonics for Solving the Incompressible Navier-Stokes Equation on the Sphere (2004)
- This work is concerned with a nonlinear Galerkin method for solving the incompressible Navier-Stokes equation on the sphere. It extends the work of Debussche, Marion,Shen, Temam et al. from one-dimensional or toroidal domains to the spherical geometry. In the first part, the method based on type 3 vector spherical harmonics is introduced and convergence is indicated. Further it is shown that the occurring coupling terms involving three vector spherical harmonics can be expressed algebraically in terms of Wigner-3j coefficients. To improve the numerical efficiency and economy we introduce an FFT based pseudo spectral algorithm for computing the Fourier coefficients of the nonlinear advection term. The resulting method scales with O(N^3), if N denotes the maximal spherical harmonic degree. The latter is demonstrated in an extensive numerical example.