Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche's coupling approach

We develop a computational model to study the interaction of a fluid with a poroelastic material. The coupling of Stokes and Biot equations represents a prototype problem for these phenomena, which feature multiple facets. On one hand it shares common traits with fluid-structure interaction. On the other hand it resembles the Stokes-Darcy coupling. For these reasons, the numerical simulation of the Stokes-Biot coupled system is a challenging task. The need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners are typical shortcomings of classical discretization methods applied to this problem. The application of loosely coupled time advancing schemes mitigates these issues because it allows to solve each equation of the system independently with respect to the others. In this work we develop and thoroughly analyze a loosely coupled scheme for Stokes-Biot equations. The scheme is based on Nitsche's method for enforcing interface conditions. Once the interface operators corresponding to the interface conditions have been defined, time lagging allows us to build up a loosely coupled scheme with good stability properties. The stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem. The error of the resulting method is also analyzed, showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes. In order to restore good approximation properties, while maintaining the computational efficiency of the loosely coupled approach, we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach. Both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand

Partitioned time discretization for atmosphere-ocean interaction

Numerical algorithms are proposed, analyzed and tested for improved efficiency and reliabil-ity of the dynamic core of climate codes. The commonly used rigid lid hypothesis is assumed,which allows instantaneous response of the interface to changes in mass. Additionally, mois-ture transport is ignored, resulting in a static interface. A central algorithmic feature is thenumerical decoupling of the atmosphere and ocean calculations by a semi-implicit treatmentof the interface data, i.e. partitioned time stepping. Algorithms are developed for simpli-fied continuum models retaining the key mathematical structure of the atmosphere-oceanequations. The work begins by studying linear parameterization of momentum flux in terms of windshear, coupling the equations. Partitioned variants of backward-Euler are developed allowinglarge time steps. Higher order accuracy is achieved by deferred correction. Adaptations aredeveloped for nonlinear coupling. Most notably an application of geometric averaging isused to retain unconditional stability. This algorithm is extended to allow different size timesteps for the subcalculations. Full numerical analyses are performed and computationalexperiments are provided. Next, heat convection is added including a nonlinear parameterization of heat flux interms of wind shear and temperature. A partitioned algorithm is developed for the atmo-sphere and ocean coupled velocity-temperature system that retains unconditional stability.Furthermore, uncertainty quantification is performed in this case due to the importance ofreliably calculating heat transport phenomena in climate modeling. Noise is introduced in two coupling parameters with an important role in stability. Numerical tests investigate thevariance in temperature, velocity and average surface temperature. Partitioned methods are highly efficient for linearly coupled 2 fluid problems. Exten-sions of these methods for nonlinear coupling where the interface data is processed properlybefore passing yield highly efficient algorithms. One reason is due to their strong stabilityproperties. Convergence also holds under time step restrictions not dependent on mesh size.It is observed that two-way coupling (requiring knowledge of both atmosphere and oceanvelocities on the interface) generates less uncertainty in the calculation of average surfacetemperature compared to one-way models (only requiring knowledge of the wind velocity)

HIGHER ACCURACY METHODS FOR FLUID FLOWS IN VARIOUS APPLICATIONS: THEORY AND IMPLEMENTATION

This dissertation contains research on several topics related to Defect-deferred correction (DDC) method applying to CFD problems. First, we want to improve the error due to temporal discretization for the problem of two convection dominated convection-diffusion problems, coupled across a joint interface. This serves as a step towards investigating an atmosphere-ocean coupling problem with the interface condition that allows for the exchange of energies between the domains. The main diffuculty is to decouple the problem in an unconditionally stable way for using legacy code for subdomains. To overcome the issue, we apply the Deferred Correction (DC) method. The DC method computes two successive approximations and we will exploit this extra flexibility by also introducing the artificial viscosity to resolve the low viscosity issue. The low viscosity issue is to lose an accuracy and a way of finding a approximate solution as a diffusion coeffiscient gets low. Even though that reduces the accuracy of the first approximation, we recover the second order accuracy in the correction step. Overall, we construct a defect and deferred correction (DDC) method. So that not only the second order accuracy in time and space is obtained but the method is also applicable to flows with low viscosity. Upon successfully completing the project in Chapter 1, we move on to implementing similar ideas for a fluid-fluid interaction problem with nonlinear interface condition; the results of this endeavor are reported in Chapter 2. In the third chapter, we represent a way of using an algorithm of an existing penalty-projection for MagnetoHydroDynamics, which allows for the usage of the less sophisticated and more computationally attractive Taylor-Hood pair of finite element spaces. We numerically show that the new modification of the method allows to get first order accuracy in time on the Taylor-Hood finite elements while the existing method would fail on it. In the fourth chapter, we apply the DC method to the magnetohydrodynamic (MHD) system written in Elsásser variables to get second order accuracy in time. We propose and analyze an algorithm based on the penalty projection with graddiv stabilized Taylor Hood solutions of Elsásser formulations