scholarly journals Weak solutions for unidirectional gradient flows: existence, uniqueness, and convergence of time discretization schemes

2021 ◽  
Vol 28 (6) ◽  
Author(s):  
Masato Kimura ◽  
Matteo Negri
Keyword(s):  
Continuous Dependence ◽  
Selection Criterion ◽  
Gradient Flow ◽  
Time Discretization ◽  
Energy Identity ◽  
Gradient Flows ◽  
Monotonicity Constraints ◽  
Discretization Schemes ◽  
Monotonicity Constraint ◽  
Power And Energy

AbstractWe consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraints. First, we provide a notion of weak solution, inspired by the theory of curves of maximal slope, and then we prove existence (employing time-discrete schemes with different implementations of the constraint), uniqueness, power and energy identity, comparison principle and continuous dependence. As a by-product, we show that the energy identity gives a selection criterion for the (non-unique) evolutions obtained by other notions of solutions. Finally, we show that for autonomous energies the evolution obtained with the monotonicity constraint actually coincides with the evolution obtained by replacing the constraint with a fixed obstacle, given by the initial datum.

From metastable to coherent sets— Time-discretization schemes

2019 ◽  
Vol 29 (1) ◽  
pp. 012101
Author(s):  
Konstantin Fackeldey ◽  
Péter Koltai ◽  
Peter Névir ◽  
Henning Rust ◽  
Axel Schild ◽  
...  
Keyword(s):  
Time Discretization ◽  
Discretization Schemes

scholarly journals Macroscopic limit of the Becker–Döring equation via gradient flows

2019 ◽  
Vol 25 ◽  
pp. 22 ◽  
Author(s):  
André Schlichting
Keyword(s):  
Initial Data ◽  
Time Scale ◽  
Continuous Dependence ◽  
Monomer Concentration ◽  
Small Cluster ◽  
Gradient Structure ◽  
Gradient Flows ◽  
Cluster Distribution ◽  
Wagner Equation

This work considers gradient structures for the Becker–Döring equation and its macroscopic limits. The result of Niethammer [J. Nonlinear Sci. 13 (2003) 115–122] is extended to prove the convergence not only for solutions of the Becker–Döring equation towards the Lifshitz–Slyozov–Wagner equation of coarsening, but also the convergence of the associated gradient structures. We establish the gradient structure of the nonlocal coarsening equation rigorously and show continuous dependence on the initial data within this framework. Further, on the considered time scale the small cluster distribution of the Becker–Döring equation follows a quasistationary distribution dictated by the monomer concentration.

TWO DISCRETIZATION SCHEMES FOR A TIME-DOMAIN DISSIPATIVE ACOUSTICS PROBLEM

2006 ◽  
Vol 16 (10) ◽  
pp. 1559-1598 ◽  
Author(s):  
ALFREDO BERMÚDEZ ◽  
RODOLFO RODRÍGUEZ ◽  
DUARTE SANTAMARINA
Keyword(s):  
Time Domain ◽  
Existence And Uniqueness ◽  
Energy Equation ◽  
Time Discretization ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Temperature Fields ◽  
Displacement Fields ◽  
Discretization Schemes ◽  
Initial Boundary Value

This paper deals with a time-domain mathematical model for dissipative acoustics and is organized as follows. First, the equations of this model are written in terms of displacement and temperature fields and an energy equation is obtained. The resulting initial-boundary value problem is written in a functional framework allowing us to prove the existence and uniqueness of solution. Next, two different time-discretization schemes are proposed, and stability and error estimates are proved for both. Finally, numerical results are reported which were obtained by combining these time-schemes with Lagrangian and Raviart–Thomas finite elements for temperature and displacement fields, respectively.

NUMERICAL GRADIENT FLOW DISCRETIZATION OF VISCOUS THIN FILMS ON CURVED GEOMETRIES

2013 ◽  
Vol 23 (05) ◽  
pp. 917-947 ◽  
Author(s):  
MARTIN RUMPF ◽  
ORESTIS VANTZOS
Keyword(s):  
Thin Films ◽  
Transport Equation ◽  
Degrees Of Freedom ◽  
Gradient Flow ◽  
Time Discretization ◽  
Time Step ◽  
Time Stepping ◽  
Exterior Calculus ◽  
Natural Time ◽  
Space Discretization

The evolution of a viscous thin film on a curved geometry is numerically approximated based on the natural time discretization of the underlying gradient flow. This discretization leads to a variational problem to be solved at each time step, which reflects the balance between the decay of the free (gravitational and surface) energy and the viscous dissipation. Both dissipation and energy are derived from a lubrication approximation for a small ratio between the characteristic film height and the characteristic length scale of the surface. The dissipation is formulated in terms of a corresponding flux field, whereas the energy primarily depends on the fluid volume per unit surface, which is a conserved quantity. These two degrees of freedom are coupled by the underlying transport equation. Hence, one is naturally led to a PDE-constrained optimization problem, where the variational time stepping problem has to be solved under the constraint described by the transport equation. For the space discretization a discrete exterior calculus approach is investigated. Various applications demonstrate the qualitative and quantitative behavior of one- and two-dimensional thin films on curved geometries.

A New Class of Time Discretization Schemes for the Solution of Nonlinear PDEs

1998 ◽  
Vol 147 (2) ◽  
pp. 362-387 ◽  
Author(s):  
Gregory Beylkin ◽  
James M. Keiser ◽  
Lev Vozovoi
Keyword(s):  
Time Discretization ◽  
Nonlinear Pdes ◽  
New Class ◽  
Discretization Schemes

scholarly journals On the Accuracy and Efficiency of Transient Spectral Element Models for Seismic Wave Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Sanna Mönkölä
Keyword(s):  
Fourth Order ◽  
Wave Equations ◽  
Seismic Waves ◽  
Time Discretization ◽  
Spectral Element ◽  
Higher Order ◽  
Time Stepping ◽  
Discretization Schemes ◽  
Elastic Wave Equations ◽  
Wave Problems

This study concentrates on transient multiphysical wave problems for simulating seismic waves. The presented models cover the coupling between elastic wave equations in solid structures and acoustic wave equations in fluids. We focus especially on the accuracy and efficiency of the numerical solution based on higher-order discretizations. The spatial discretization is performed by the spectral element method. For time discretization we compare three different schemes. The efficiency of the higher-order time discretization schemes depends on several factors which we discuss by presenting numerical experiments with the fourth-order Runge-Kutta and the fourth-order Adams-Bashforth time-stepping. We generate a synthetic seismogram and demonstrate its function by a numerical simulation.

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