Computational Mathematics, Publications

We present a high-order difference method for problems in elastodynamics involving

the interaction of waves with highly nonlinear frictional interfaces. We restrict our

attention to two-dimensional antiplane problems involving deformation in only one direction.

Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity

across it are of a form closely related to those used in earthquake rupture models and

other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators

and weak enforcement of boundary and interface conditions, a strictly stable method

is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated

in terms of characteristic variables, as opposed to the physical variables in terms of

which they are more naturally stated, the semi-discretized system of equations can become

extremely stiff, preventing efficient solution using explicit time integrators.

The use of SBP operators also provides a rigorously defined energy balance for the discretized

problem that, as the mesh is refined, approaches the exact energy balance in the

continuous problem. This enables one to investigate earthquake energetics, for example the

efficiency with which elastic strain energy released during rupture is converted to radiated

energy carried by seismic waves, rather than dissipated by frictional sliding of the fault.

These theoretical results are confirmed by several numerical tests in both one and two dimensions

demonstrating the computational efficiency, the high-order convergence rate of

the method, the benefits of using strictly stable numerical methods for long time integration,

and the accuracy of the energy balance.

The problem of computing the best rank-(p,q,r) approximation of a third order tensor is considered. First the problem is reformulated as a maximization problem on a product of three Grassmann manifolds. Then expressions for the gradient and the Hessian are derived in a local coordinate system at a stationary point, and conditions for a local maximum are given. A first order perturbation analysis is performed using the Grassmann manifold framework. The analysis is illustrated in a few examples, and it is shown that the perturbation theory for the singular value decomposition is a special case of the tensor theory.

In this paper we prove stability of Robin solid wall boundary conditions for the compressible Navier–Stokes equations. Applications include the no-slip boundary conditions with prescribed temperature or temperature gradient and the first order slip-flow boundary conditions. The formulation is uniform and the transitions between different boundary conditions are done by a change of parameters. We give different sharp energy estimates depending on the choice of parameters.

The discretization is done using finite differences on Summation-By-Parts form with weak boundary conditions using the Simultaneous Approximation Term. We verify convergence by the method of manufactured solutions and show computations of flows ranging from no-slip to almost full slip.

Unstructured grid methods are well suited for earthquake problems in complex geometries, such as non-planar and branching faults. Unfortunately they are indefficient in comparsion wiht high-order finite differences. With the use of summation-by-parts (SBP) operators and the SAT penalty method (simultaneous approximation term) it is possible to couple unstructured finite volume methods with high-order finite difference methods in an accurate and stable way. The couple method is more efficient than the unstructured method alone. Another advantage of the SBP and SAT method is that it is possible to prove strict stability, meaning that the semi-discrete solution dissipates energy at a slightly faster rate than the continuous solution so that the error remains bounded in time, which is particulary useful for long time computations.

We develop a stable and high-order accurate finite difference method for problems in earthquake rupture dynamics capable of handling complex geometries and multiple faults. The bulk material is an isotropic elastic solid cut by preexisting fault interfaces. The fields across the interfaces are related through friction laws which depend on the sliding velocity, tractions acting on the interface, and state variables which evolve according to ordinary differential equations involving local fields.

The method is based on summation-by-parts finite difference operators with irregular geometries handled through coordinate transforms and multi-block meshes. Boundary conditions as well as block interface conditions (whether frictional or otherwise) are enforced weakly through the simultaneous approximation term method, resulting in a provably stable discretization.

The theoretical accuracy and stability results are confirmed with the method of manufactured solutions. The practical benefits of the new methodology are illustrated in a simulation of a subduction zone megathrust earthquake, a challenging application problem involving complex free-surface topography, nonplanar faults, and varying material properties.

We study the influence of different implementations of no-slip solid wall boundary conditions on the convergence to steady-state of the Navier-Stokes equations. The various approaches are investigated using the energy method and an eigenvalue analysis. It is shown that the weak implementation is superior and enhances the convergence to steady-state for coarse meshes. It is also demonstrated that all the stable approaches produce the same convergence rate as the mesh size goes to zero. The numerical results obtained by using a fully nonlinear finite volume solver support the theoretical findings from the linear analysis.

We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators.

The use of SBP operators also provides a rigorously defined energy balance for the discretized problem that, as the mesh is refined, approaches the exact energy balance in the continuous problem. This enables one to investigate earthquake energetics, for example the efficiency with which elastic strain energy released during rupture is converted to radiated energy carried by seismic waves, rather than dissipated by frictional sliding of the fault. These theoretical results are confirmed by several numerical tests in both one and two dimensions demonstrating the computational efficiency, the high-order convergence rate of the method, the benefits of using strictly stable numerical methods for long time integration, and the accuracy of the energy balance.

Using fnite difference methods for partial differential equations, this thesis focuses on the problem of connecting overlapping solution domains in the context of a frst order hyperbolic problem. Especially the stability properties of such constructions is studied, and a stable general implementation of the the test problem is proposed. However, no energy estimate could be found, and indeed proven not to exist in the natural norm. Finally, an example is also put forward where the interface conditions derived are, for stability considerations, incompatible with the boundary conditions in a coupled system of hyperbolic equations.