23 research outputs found
Summation-By-Parts Operators and High-Order Quadrature
Summation-by-parts (SBP) operators are finite-difference operators that mimic
integration by parts. This property can be useful in constructing energy-stable
discretizations of partial differential vequations. SBP operators are defined
by a weight matrix and a difference operator, with the latter designed to
approximate to a specified order of accuracy. The accuracy of the weight
matrix as a quadrature rule is not explicitly part of the SBP definition. We
show that SBP weight matrices are related to trapezoid rules with end
corrections whose accuracy matches the corresponding difference operator at
internal nodes. The accuracy of SBP quadrature extends to curvilinear domains
provided the Jacobian is approximated with the same SBP operator used for the
quadrature. This quadrature has significant implications for SBP-based
discretizations; for example, the discrete norm accurately approximates the
norm for functions, and multi-dimensional SBP discretizations
accurately mimic the divergence theorem.Comment: 18 pages, 3 figure
Quadrature Rules on Triangles and Tetrahedra for Multidimensional Summation-By-Parts Operators
Multidimensional diagonal-norm summation-by-parts (SBP) operators with
collocated volume and facet nodes, known as diagonal- operators,
are attractive for entropy-stable discretizations from an efficiency
standpoint. However, there is a limited number of such operators, and those
currently in existence often have a relatively high node count for a given
polynomial order due to a scarcity of suitable quadrature rules. We present
several new symmetric positive-weight quadrature rules on triangles and
tetrahedra that are suitable for construction of diagonal- SBP
operators. For triangles, quadrature rules of degree one through twenty with
facet nodes that correspond to the Legendre-Gauss-Lobatto (LGL) and
Legendre-Gauss (LG) quadrature rules are derived. For tetrahedra, quadrature
rules of degree one through ten are presented along with the corresponding
facet quadrature rules. All of the quadrature rules are provided in a
supplementary data repository. The quadrature rules are used to construct novel
SBP diagonal- operators, whose accuracy and maximum timestep
restrictions are studied numerically.Comment: 11 pages, 1 figur
Modeling Anisotropy in the Earthquake Cycle: a Numerical Multiscale Model
We have developed a methodology for incorporating and studying the effects of anisotropy when simulating the full earthquake cycle. The method is developed for a vertical strike-slip fault in two-dimensions, with antiplane motion. Inertial terms are dropped from the elastic anisotropic wave equation to obtain a steady state problem. This resulting equilibrium equation is discretized with a finite difference method. A nonlinear rate-and-state friction law is enforced at the fault. Time stepping is adaptive to capture highly varying time scales, and as such is able to produce self-consistent initial conditions
Review of Summation-by-parts schemes for initial-boundary-value problems
High-order finite difference methods are efficient, easy to program, scales
well in multiple dimensions and can be modified locally for various reasons
(such as shock treatment for example). The main drawback have been the
complicated and sometimes even mysterious stability treatment at boundaries and
interfaces required for a stable scheme. The research on summation-by-parts
operators and weak boundary conditions during the last 20 years have removed
this drawback and now reached a mature state. It is now possible to construct
stable and high order accurate multi-block finite difference schemes in a
systematic building-block-like manner. In this paper we will review this
development, point out the main contributions and speculate about the next
lines of research in this area
An Improved High Order Finite Difference Method for Non-conforming Grid Interfaces for the Wave Equation
This paper presents an extension of a recently developed high order finite difference method for the wave equation on a grid with non-conforming interfaces. The stability proof of the existing methods relies on the interpolation operators being norm-contracting, which is satisfied by the second and fourth order operators, but not by the sixth order operator. We construct new penalty terms to impose interface conditions such that the stability proof does not require the norm-contracting condition. As a consequence, the sixth order accurate scheme is also provably stable. Numerical experiments demonstrate the improved stability and accuracy property
An Improved High Order Finite Difference Method for Non-conforming Grid Interfaces for the Wave Equation
This paper presents an extension of a recently developed high order finite difference method for the wave equation on a grid with non-conforming interfaces. The stability proof of the existing methods relies on the interpolation operators being norm-contracting, which is satisfied by the second and fourth order operators, but not by the sixth order operator. We construct new penalty terms to impose interface conditions such that the stability proof does not require the norm-contracting condition. As a consequence, the sixth order accurate scheme is also provably stable. Numerical experiments demonstrate the improved stability and accuracy property