scholarly journals On the boundedness stepsizes-coefficients of A-BDF methods

2021 ◽  
Vol 7 (2) ◽  
pp. 1562-1579
Author(s):  
Dumitru Baleanu ◽  
◽  
Mohammad Mehdizadeh Khalsaraei ◽  
Ali Shokri ◽  
Kamal Kaveh ◽  
...  
Keyword(s):  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Physical Phenomenon ◽  
Diffusion Problem ◽  
Biological Species ◽  
Linear Multistep Methods ◽  
Qualitative Properties ◽  
Physical Constraints ◽  
Bdf Methods ◽  
Bdf Method

<abstract><p>Physical constraints must be taken into account in solving partial differential equations (PDEs) in modeling physical phenomenon time evolution of chemical or biological species. In other words, numerical schemes ought to be devised in a way that numerical results may have the same qualitative properties as those of the theoretical results. Methods with monotonicity preserving property possess a qualitative feature that renders them practically proper for solving hyperbolic systems. The need for monotonicity signifies the essential boundedness properties necessary for the numerical methods. That said, for many linear multistep methods (LMMs), the monotonicity demands are violated. Therefore, it cannot be concluded that the total variations of those methods are bounded. This paper investigates monotonicity, especially emphasizing the stepsize restrictions for boundedness of A-BDF methods as a subclass of LMMs. A-stable methods can often be effectively used for stiff ODEs, but may prove inefficient in hyperbolic equations with stiff source terms. Numerical experiments show that if we apply the A-BDF method to Sod's problem, the numerical solution for the density is sharp without spurious oscillations. Also, application of the A-BDF method to the discontinuous diffusion problem is free of temporal oscillations and negative values near the discontinuous points while the SSP RK2 method does not have such properties.</p></abstract>

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.

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.

ON A MODEL FOR THE PROPAGATION OF ISOTOPIC DISEQUILIBRIUM BY DIFFUSION

2009 ◽  
Vol 19 (08) ◽  
pp. 1277-1294 ◽  
Author(s):  
E. COMPARINI ◽  
R. DAL PASSO ◽  
C. PESCATORE ◽  
M. UGHI
Keyword(s):  
Parabolic Equation ◽  
Chemical Element ◽  
Chemical Potential ◽  
Hyperbolic Equations ◽  
Total Concentration ◽  
The Other ◽  
Qualitative Properties ◽  
Used Nuclear Fuel ◽  
Solvent Molecules ◽  
Isotopic Disequilibrium

We consider a model for the distribution of radionuclides in the ground water around a deep repository for used nuclear fuel, based on the assumption that different isotopes of the same chemical element A contribute jointly to the chemical potential of A. In this hypothesis, the total flux Ji of a particular isotope Ai of an element A has two components, one due to the interaction of Ai with the solvent molecules B, the other with the kin isotopes. We study some qualitative properties of the solution in the physically relevant assumption that the first of these components is negligible. In this assumption the problem reduces to a parabolic equation for the total concentration of the element A, possibly coupled with hyperbolic equations for the concentrations of the single isotopes.

scholarly journals Coupling techniques for nonlinear hyperbolic equations. I Self-similar diffusion for thin interfaces

2011 ◽  
Vol 141 (5) ◽  
pp. 921-956 ◽  
Author(s):  
Benjamin Boutin ◽  
Frédéric Coquel ◽  
Philippe G. LeFloch
Keyword(s):  
Riemann Problem ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Resonance Effect ◽  
Wave Interaction ◽  
Technical Difficulty ◽  
Nonlinear Hyperbolic Equations ◽  
Global Hyperbolicity ◽  
Nonlinear Hyperbolic System ◽  
Self Similar

We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce an augmented formulation that allows for the modelling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial-value problem, and these solutions need to be supplemented with further admissibility conditions. This paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions that apply to resonant wave patterns.

Diagnostics of a gradient catastrophe for a class of quasilinear hyperbolic systems

2017 ◽  
Vol 32 (1) ◽  
Author(s):  
Eugene V. Chizhonkov
Keyword(s):  
Burgers Equation ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Axially Symmetric ◽  
Two Stage ◽  
One Dimensional ◽  
Quasilinear Hyperbolic Systems ◽  
Quasilinear Systems ◽  
Stage Analysis ◽  
Plane Oscillations

AbstractA two-stage analysis do detect the appearance of a gradient catastrophe of the solution is proposed for quasilinear systems of hyperbolic equations of special form. Applications of the first stage are considered for the following simple cases: scalar Burgers’ equation and a quasi-orthogonal system generalizing it. The entire two-stage analysis is applied to systems of equations describing one-dimensional electron oscillations in plasma, namely, plane oscillations in the relativistic and non-relativistic cases and also axially symmetric non-relativistic cylindrical oscillations.

scholarly journals Linear Multistep Methods for Impulsive Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
X. Liu ◽  
M. H. Song ◽  
M. Z. Liu
Keyword(s):  
Differential Equations ◽  
Numerical Experiments ◽  
Multistep Methods ◽  
Impulsive Differential Equations ◽  
Multistep Method ◽  
Linear Multistep Methods ◽  
Linear Multistep Method ◽  
Convergence And Stability ◽  
Improved Methods ◽  
Bdf Method

This paper deals with the convergence and stability of linear multistep methods for impulsive differential equations. Numerical experiments demonstrate that both the mid-point rule and two-step BDF method are of orderp=0when applied to impulsive differential equations. An improved linear multistep method is proposed. Convergence and stability conditions of the improved methods are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.

scholarly journals Linear Hyperbolic Systems on Networks: well-posedness and qualitative properties

2020 ◽  
Author(s):  
Marjeta Kramar ◽  
Delio Mugnolo ◽  
Serge Nicaise
Keyword(s):  
Boundary Conditions ◽  
Evolution Equations ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Order Reduction ◽  
Well Posedness ◽  
Qualitative Properties ◽  
Local Boundary ◽  
First Order ◽  
Necessary And Sufficient

We study hyperbolic systems of one - dimensional partial differential equations under general , possibly non-local boundary conditions. A large class of evolution equations, either on individual 1- dimensional intervals or on general networks , can be reformulated in our rather flexible formalism , which generalizes the classical technique of first - order reduction . We study forward and backward well - posedness ; furthermore , we provide necessary and sufficient conditions on both the boundary conditions and the coefficients arising in the first - order reduction for a given subset of the relevant ambient space to be invariant under the flow that governs the system. Several examples are studied . p, li { white-space: pre-wrap; }

GRAPH SOLUTIONS OF NONLINEAR HYPERBOLIC SYSTEMS

2004 ◽  
Vol 01 (04) ◽  
pp. 643-689 ◽  
Author(s):  
PHILIPPE G. LEFLOCH
Keyword(s):  
Cauchy Problem ◽  
Traveling Wave ◽  
Space Dimension ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Small Scale ◽  
Geometric Framework ◽  
Uniform Distance ◽  
The Cauchy Problem ◽  
Nonlinear Hyperbolic Systems

For nonlinear hyperbolic systems of partial differential equations in one-space dimension (in either conservative or non-conservative form) we introduce a geometric framework in which solutions are sought as (continuous) parametrized graphs(t,s) ↦ (X,U)(t,s) satisfying ∂sX ≥ 0, rather than (discontinuous) functions (t,x) ↦ u(t,x). On one hand, we generalize an idea by Dal Maso, LeFloch, and Murat who used a family of traveling wave profiles to define non-conservative products, and we define the notion of graph solution subordinate to a family of Riemann graphs. The latter naturally encodes the graph of the solution to the Riemann problem, which should be determined from an augmented model taking into account small-scale physics and providing an internal structure to the shock waves. In a second definition, we write an evolution equation on the graphs directly and we introduce the notion of graph solution subordinate to a diffusion matrix, which merges together the hyperbolic equations (in the "non-vertical" parts of the graphs) with the traveling wave equation of the augmented model (in the "vertical" parts). We consider the Cauchy problem within the class of graph solutions. The graph solution to the Cauchy problem is constructed by completion of the discontinuities of the entropy solution. The uniqueness is established by applying a general uniqueness theorem due to Baiti, LeFloch, and Piccoli. The proposed geometric framework illustrates the importance of the uniform distance between graphs to deal with solutions of nonlinear hyperbolic problems.

The numerical solution of hyperbolic systems of partial differential equations in three independent variables

1960 ◽  
Vol 255 (1281) ◽  
pp. 232-252 ◽  
Keyword(s):  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Original Method ◽  
Plane Motion ◽  
Step Size ◽  
Independent Variables ◽  
Step Procedure ◽  
Dependent Variables ◽  
Derivatives Of ◽  
Unknown Point

An original method of integration is described for quasi-linear hyperbolic equations in three independent variables. The solution is constructed by means of a step-by-step procedure, employing difference relations along four bicharacteristics and one time-like ordinary curve through each point. From these difference relations the derivatives of the dependent variables at the unknown point are eliminated. The solution at any point can then be com­puted, with an error proportional to the step size cubed, without referring to conditions outside its domain of dependence. The application of the method to the systems of equations governing unsteady plane motion and steady supersonic flow of an inviscid, non-conducting fluid is discussed in detail. As an example of the use of the method, the flow over a particular delta-shaped body has been computed.

scholarly journals Duality in the optimal control for damped hyperbolic systems with positive control

2003 ◽  
Vol 2003 (27) ◽  
pp. 1703-1714
Author(s):  
Mi Jin Lee ◽  
Jong Yeoul Park ◽  
Young Chel Kwon
Keyword(s):  
Optimal Control ◽  
Duality Theory ◽  
Duality Theorem ◽  
Hyperbolic Equations ◽  
Hyperbolic Systems ◽  
Positive Control ◽  
Cost Functions ◽  
Convex Cost ◽  
Cost Functionals ◽  
Positive Controls

We study the duality theory for damped hyperbolic equations. These systems have positive controls and convex cost functionals. Our main results lie in the application of duality theorem, that is,inf J=sup K, on various cost functions.

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