Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations

We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerate-elliptic operators of the kind described in our article appear in a diverse range of applications, including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance, generators of diffusion processes arising in mathematical biology, and the study of porous media.Comment: 58 pages, 1 figure. To appear in the Journal of Differential Equations. Incorporates final galley proof corrections corresponding to published versio

Solving seismic wave propagation in elastic media using the matrix exponential approach

Three numerical algorithms are proposed to solve the time-dependent elastodynamic equations in elastic solids. All algorithms are based on approximating the solution of the equations, which can be written as a matrix exponential. By approximating the matrix exponential with a product formula, an unconditionally stable algorithm is derived that conserves the total elastic energy density. By expanding the matrix exponential in Chebyshev polynomials for a specific time instance, a so-called ``one-step'' algorithm is constructed that is very accurate with respect to the time integration. By formulating the conventional velocity-stress finite-difference time-domain algorithm (VS-FDTD) in matrix exponential form, the staggered-in-time nature can be removed by a small modification, and higher order in time algorithms can be easily derived. For two different seismic events the accuracy of the algorithms is studied and compared with the result obtained by using the conventional VS-FDTD algorithm.Comment: 13 pages revtex, 6 figures, 2 table

Continuous Symmetries of Difference Equations

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl