63 research outputs found
Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods with High Linear Order
When evolving in time the solution of a hyperbolic partial differential
equation, it is often desirable to use high order strong stability preserving
(SSP) time discretizations. These time discretizations preserve the
monotonicity properties satisfied by the spatial discretization when coupled
with the first order forward Euler, under a certain time-step restriction.
While the allowable time-step depends on both the spatial and temporal
discretizations, the contribution of the temporal discretization can be
isolated by taking the ratio of the allowable time-step of the high order
method to the forward Euler time-step. This ratio is called the strong
stability coefficient. The search for high order strong stability time-stepping
methods with high order and large allowable time-step had been an active area
of research. It is known that implicit SSP Runge-Kutta methods exist only up to
sixth order. However, if we restrict ourselves to solving only linear
autonomous problems, the order conditions simplify and we can find implicit SSP
Runge-Kutta methods of any linear order. In the current work we aim to find
very high linear order implicit SSP Runge-Kutta methods that are optimal in
terms of allowable time-step. Next, we formulate an optimization problem for
implicit-explicit (IMEX) SSP Runge-Kutta methods and find implicit methods with
large linear stability regions that pair with known explicit SSP Runge-Kutta
methods of orders plin=3,4,6 as well as optimized IMEX SSP Runge-Kutta pairs
that have high linear order and nonlinear orders p=2,3,4. These methods are
then tested on sample problems to verify order of convergence and to
demonstrate the sharpness of the SSP coefficient and the typical behavior of
these methods on test problems
Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge--Kutta methods
We construct a family of embedded pairs for optimal strong stability
preserving explicit Runge-Kutta methods of order to be used
to obtain numerical solution of spatially discretized hyperbolic PDEs. In this
construction, the goals include non-defective methods, large region of absolute
stability, and optimal error measurement as defined in [5,19]. The new family
of embedded pairs offer the ability for strong stability preserving (SSP)
methods to adapt by varying the step-size based on the local error estimation
while maintaining their inherent nonlinear stability properties. Through
several numerical experiments, we assess the overall effectiveness in terms of
precision versus work while also taking into consideration accuracy and
stability.Comment: 22 pages, 49 figure
Local-in-time structure-preserving finite-element schemes for the Euler-Poisson equations
We discuss structure-preserving numerical discretizations for repulsive and
attractive Euler-Poisson equations that find applications in fluid-plasma and
self-gravitation modeling, respectively. The scheme is fully discrete and
structure preserving in the sense that it maintains a discrete energy law, as
well as hyperbolic invariant domain properties, such as positivity of the
density and a minimum principle of the specific entropy. A detailed discussion
of algorithmic details is given, as well as proofs of the claimed properties.
We present computational experiments corroborating our analytical findings and
demonstrating the computational capabilities of the scheme
Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements
This work extends the flux-corrected transport (FCT) methodology to arbitrary-order continuous finite element discretizations
of scalar conservation laws on simplex meshes. Using Bernstein polynomials as local basis functions, we
constrain the total variation of the numerical solution by imposing local discrete maximum principles on the Bézier
net. The design of accuracy-preserving FCT schemes for high order Bernstein-Bézier finite elements requires the development
of new algorithms and/or generalization of limiting techniques tailored for linear and multilinear Lagrange
elements. In this paper, we propose (i) a new discrete upwinding strategy leading to local extremum bounded low
order approximations with compact stencils, (ii) a high order stabilization operator based on gradient recovery, and
(iii) new localized limiting techniques for antidi usive element contributions. The optional use of a smoothness indicator,
based on a second derivative test, makes it possible to potentially avoid unnecessary limiting at smooth extrema
and achieve optimal convergence rates for problems with smooth solutions. The accuracy of the proposed schemes is
assessed in numerical studies for the linear transport equation in 1D and 2D
Recommended from our members
Inexact Newton Dogleg Methods
The dogleg method is a classical trust-region technique for globalizing Newton\u27s method. While it is widely used in optimization, including large-scale optimization via truncated-Newton approaches, its implementation in general inexact Newton methods for systems of nonlinear equations can be problematic. In this paper, we first outline a very general dogleg method suitable for the general inexact Newton context and provide a global convergence analysis for it. We then discuss certain issues that may arise with the standard dogleg implementational strategy and propose modified strategies that address them. Newton-Krylov methods have provided important motivation for this work, and we conclude with a report on numerical experiments involving a Newton-GMRES dogleg method applied to benchmark CFD problems
- …