31 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
High-order accurate difference schemes for the Hodgkin-Huxley equations
A novel approach for simulating potential propagation in neuronal branches
with high accuracy is developed. The method relies on high-order accurate
difference schemes using the Summation-By-Parts operators with weak boundary
and interface conditions applied to the Hodgkin-Huxley equations. This work is
the first demonstrating high accuracy for that equation. Several boundary
conditions are considered including the non-standard one accounting for the
soma presence, which is characterized by its own partial differential equation.
Well-posedness for the continuous problem as well as stability of the discrete
approximation is proved for all the boundary conditions. Gains in terms of CPU
times are observed when high-order operators are used, demonstrating the
advantage of the high-order schemes for simulating potential propagation in
large neuronal trees
Higher order finite difference schemes for the magnetic induction equations
We describe high order accurate and stable finite difference schemes for the
initial-boundary value problem associated with the magnetic induction
equations. These equations model the evolution of a magnetic field due to a
given velocity field. The finite difference schemes are based on Summation by
Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation
Term (SAT) technique for imposing boundary conditions. We present various
numerical experiments that demonstrate both the stability as well as high order
of accuracy of the schemes.Comment: 20 page
Adjoint-based sensitivity analysis of ignition in a turbulent reactive shear layer
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143014/1/6.2017-0846.pd
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
Direct numerical simulation of noise suppression by water injection in high-speed flows
Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143117/1/6.2017-1700.pd
Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach
In this paper, we consider Burgers' equation with uncertain boundary and
initial conditions. The polynomial chaos (PC) approach yields a hyperbolic
system of deterministic equations, which can be solved by several numerical
methods. Here, we apply the correction procedure via reconstruction (CPR) using
summation-by-parts operators. We focus especially on stability, which is proven
for CPR methods and the systems arising from the PC approach. Due to the usage
of split-forms, the major challenge is to construct entropy stable numerical
fluxes. For the first time, such numerical fluxes are constructed for all
systems resulting from the PC approach for Burgers' equation. In numerical
tests, we verify our results and show also the advantage of the given ansatz
using CPR methods. Moreover, one of the simulations, i.e. Burgers' equation
equipped with an initial shock, demonstrates quite fascinating observations.
The behaviour of the numerical solutions from several methods (finite volume,
finite difference, CPR) differ significantly from each other. Through careful
investigations, we conclude that the reason for this is the high sensitivity of
the system to varying dissipation. Furthermore, it should be stressed that the
system is not strictly hyperbolic with genuinely nonlinear or linearly
degenerate fields
Summation-by-parts operators for general function spaces: The second derivative
Many applications rely on solving time-dependent partial differential
equations (PDEs) that include second derivatives. Summation-by-parts (SBP)
operators are crucial for developing stable, high-order accurate numerical
methodologies for such problems. Conventionally, SBP operators are tailored to
the assumption that polynomials accurately approximate the solution, and SBP
operators should thus be exact for them. However, this assumption falls short
for a range of problems for which other approximation spaces are better suited.
We recently addressed this issue and developed a theory for first-derivative
SBP operators based on general function spaces, coined function-space SBP
(FSBP) operators. In this paper, we extend the innovation of FSBP operators to
accommodate second derivatives. The developed second-derivative FSBP operators
maintain the desired mimetic properties of existing polynomial SBP operators
while allowing for greater flexibility by being applicable to a broader range
of function spaces. We establish the existence of these operators and detail a
straightforward methodology for constructing them. By exploring various
function spaces, including trigonometric, exponential, and radial basis
functions, we illustrate the versatility of our approach. We showcase the
superior performance of these non-polynomial FSBP operators over traditional
polynomial-based operators for a suite of one- and two-dimensional problems,
encompassing a boundary layer problem and the viscous Burgers' equation. The
work presented here opens up possibilities for using second-derivative SBP
operators based on suitable function spaces, paving the way for a wide range of
applications in the future.Comment: 20 pages, 7 figure