scholarly journals Structure aware Runge–Kutta time stepping for spacetime tents

2020 ◽  
Vol 1 (4) ◽  
Author(s):  
Jay Gopalakrishnan ◽  
Joachim Schöberl ◽  
Christoph Wintersteiger
Keyword(s):  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Convergence Properties ◽  
New Class ◽  
Time Stepping ◽  
New Methods ◽  
Optimal Convergence Rates ◽  
Runge Kutta ◽  
Discrete Stability

Abstract We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge–Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations.

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

New Methods for Analyzing Water Pollutants

1982 ◽  
Vol 14 (4-5) ◽  
pp. 59-71 ◽  
Author(s):  
L H Keith ◽  
R C Hall ◽  
R C Hanisch ◽  
R G Landolt ◽  
J E Henderson
Keyword(s):  
Amino Acids ◽  
Gas Chromatography ◽  
Two Dimensional ◽  
Gas Chromatograph ◽  
New Class ◽  
New Methods ◽  
System A ◽  
Drinking Waters ◽  
Computer Controlled ◽  
Nitrogen Containing Compounds

Two new methods have been developed to analyze for organic pollutants in water. The first, two-dimensional gas chromatography, using post detector peak recycling (PDPR), involves the use of a computer-controlled gas Chromatograph to selectively trap compounds of interest and rechromatograph them on a second column, recycling them through the same detector again. The second employs a new detector system, a thermally modulated electron capture detector (TMECD). Both methods were used to demonstrate their utility by applying them to the analysis of a new class of potentially ubiquitous anthropoaqueous pollutants in drinking waters- -haloacetonitriles. These newly identified compounds are produced from certain amino acids and other nitrogen-containing compounds reacting with chlorine during the disinfection stage of treatment.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

scholarly journals Krylov SSP Integrating Factor Runge–Kutta WENO Methods

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1483
Author(s):  
Shanqin Chen
Keyword(s):  
Large Time ◽  
Hyperbolic Equations ◽  
Strong Stability ◽  
Matrix Exponential ◽  
Linear Component ◽  
Nonlinear Hyperbolic Equations ◽  
Time Step ◽  
Integrating Factor ◽  
The Matrix ◽  
Runge Kutta

Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

scholarly journals Multigrid and Runge-Kutta time stepping applied to the uniformly non-oscillatory scheme for conservation laws

1991 ◽  
Vol 25 (3) ◽  
pp. 243-263 ◽  
Author(s):  
J. W. van der Burg ◽  
J. G. M. Kuerten ◽  
P. J. Zandbergen
Keyword(s):  
Conservation Laws ◽  
Time Stepping ◽  
Runge Kutta

A new class of isotropic generators for developing 6-DOF isotropic manipulators

Robotica ◽  
2008 ◽  
Vol 26 (5) ◽  
pp. 619-625 ◽  
Author(s):  
K. Y. Tsai ◽  
T. K. Lee ◽  
Y. S. Jang
Keyword(s):  
Reference Point ◽  
Parallel Manipulators ◽  
Geometric Constraint ◽  
Kinematic Chains ◽  
New Class ◽  
New Methods ◽  
Center Point ◽  
Different Types ◽  
Different Shapes ◽  
Straight Lines

SUMMARYDeveloping 6-DOF isotropic manipulators using isotropic generators is simple and efficient, and isotropic generators can be employed to develop serial, redundant, or parallel isotropic manipulators. An isotropic generator consists of a reference point and six straight lines. The existing generators, however, have one common geometric constraint: the reference point is equidistant from the six straight lines. Some practical isotropic designs might not be obtained due to this constraint. This paper proposes methods for developing new isotropic generators. The generators thus developed are not subject to the constraint, and the new methods allow us to specify the location of the tool center point, the size of the platform or the base, or the shape of isotropic parallel manipulators. Many new generators are presented to develop 6-DOF parallel manipulators with different shapes or different types of kinematic chains.

Time-periodic solutions of symmetric hyperbolic systems

2020 ◽  
Vol 17 (04) ◽  
pp. 707-726
Author(s):  
Masashi Ohnawa ◽  
Masahiro Suzuki
Keyword(s):  
Periodic Solutions ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Initial Boundary ◽  
Unique Existence ◽  
Periodic Problems ◽  
Time Periodic Solutions ◽  
External Forces ◽  
Initial Boundary Value ◽  
Time Periodic

We prove the unique existence of time-periodic solutions to general hyperbolic equations with periodic external forces autonomous or nonautonomous over a domain bounded by two parallel planes, provided that all the characteristics with respect to the direction normal to the planes have the same sign. It is also shown that global-in-time solutions to initial-boundary value problems coincide with the solutions to corresponding time-periodic problems after a finite time. We devote one section to the reformulation of several realistic problems and see our results have wide applicability.

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