Numerical study of blow-up and dispersive shocks in solutions to generalized Korteweg–de Vries equations

2015 ◽  
Vol 304-305 ◽  
pp. 52-78 ◽  
Author(s):  
C. Klein ◽  
R. Peter
Keyword(s):  
Blow Up ◽  
Numerical Study ◽  
De Vries

Transcritical flow over two obstacles: forced Korteweg–de Vries framework

2016 ◽  
Vol 809 ◽  
pp. 918-940 ◽  
Author(s):  
Roger H. J. Grimshaw ◽  
Montri Maleewong
Keyword(s):  
Froude Number ◽  
Numerical Study ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Control Flow ◽  
Main Concern ◽  
Undular Bore ◽  
Second Stage ◽  
Third Stage ◽  
De Vries

We consider free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation in a suite of numerical simulations. Our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. The flow behaviour can be characterised by the Froude number and the maximum heights of the obstacles. In the transcritical regime at early times, undular bores are produced upstream and downstream of each obstacle. Our main aim is to describe the interaction of these undular bores between the obstacles, and to find the outcome at very large times. We find that the flow development can be defined in three stages. The first stage is described by the well-known development of undular bores upstream and downstream of each obstacle. The second stage is the interaction between the undular bore moving downstream from the first obstacle and the undular bore moving upstream from the second obstacle. The third stage is the very large time evolution of this interaction, when one of the obstacles controls criticality. For equal obstacle heights, our analytical and numerical results indicate that either one of the obstacles can control flow criticality, that being the first obstacle when the flow is slightly subcritical and the second obstacle otherwise. For unequal obstacle heights the larger obstacle controls criticality. The results obtained here complement a recent numerical study using the fully nonlinear, but non-dispersive, shallow water equations.

scholarly journals Numerical study of blow-up and stability of line solitons for the Novikov–Veselov equation

Nonlinearity ◽  
2017 ◽  
Vol 30 (7) ◽  
pp. 2566-2591 ◽  
Author(s):  
Anna Kazeykina ◽  
Christian Klein
Keyword(s):  
Blow Up ◽  
Numerical Study

Focusing blow-up for quasilinear parabolic equations

1998 ◽  
Vol 128 (5) ◽  
pp. 965-992 ◽  
Author(s):  
C. J. Budd ◽  
V. A. Galaktionov ◽  
Jianping Chen
Keyword(s):  
Blow Up ◽  
Numerical Study ◽  
Quasilinear Parabolic Equation ◽  
Reaction Diffusion ◽  
Comparison Method ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Parabolic Equations ◽  
General Behaviour ◽  
Blowing Up ◽  
Quasilinear Parabolic

We study the behaviour of the non-negative blowing up solutions to the quasilinear parabolic equation with a typical reaction–diffusion right-hand side and with a singularity in the space variable which takes the formwhere m ≧ 1, p > 1 are arbitrary constants, in the critical exponent case q = (p–1)/m > 0. We impose zero Dirichlet boundary conditions at the singular point x = 0 and at x = 1, and take large initial data. For a class of ‘concave’ initial functions, we prove focusing at the origin of the solutions as t approaches the blow-up time T in the sense that x = 0 belongs to the blow-up set. The proof is based on an application of the intersection comparison method with an explicit ‘separable’ solution which has the same blow-up time as u. The method has a natural generalisation to the case of more general nonlinearities in the equation. A description of different fine structures of blow-up patterns in the semilinear case m = 1 and in the quasilinear one m > 1 is also presented. A numerical study of the semilinear equation is also made using an adaptive collocation method. This is shown to give very close agreement with the fine structure predicted and allows us to make some conjectures about the general behaviour.

scholarly journals Numerical study of fractional nonlinear Schrödinger equations

2014 ◽  
Vol 470 (2172) ◽  
pp. 20140364 ◽  
Author(s):  
Christian Klein ◽  
Christof Sparber ◽  
Peter Markowich
Keyword(s):  
Fractional Laplacian ◽  
Blow Up ◽  
Numerical Study ◽  
One Dimensional ◽  
Time Dynamics ◽  
Nonlinear Schrödinger ◽  
Stability And Instability ◽  
Long Time ◽  
The Stability ◽  
Fractional Nonlinear Schrödinger Equation

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.

scholarly journals FULLY DISCRETE ADAPTIVE METHODS FOR A BLOW-UP PROBLEM

2004 ◽  
Vol 14 (10) ◽  
pp. 1425-1450 ◽  
Author(s):  
CRISTINA BRÄNDLE ◽  
PABLO GROISMAN ◽  
JULIO D. ROSSI
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Blow Up ◽  
Numerical Study ◽  
Adaptive Methods ◽  
Fully Discrete ◽  
Adaptive Procedures ◽  
Exact Asymptotic Behavior ◽  
Blow Up Rate ◽  
Continuous Problem

We present adaptive procedures in space and time for the numerical study of positive solutions to the following problem: [Formula: see text] with p,m>0. We describe how to perform adaptive methods in order to reproduce the exact asymptotic behavior (the blow-up rate and the blow-up set) of the continuous problem.

NUMERICAL STUDY OF ESTIMATING THE BLOW-UP TIME OF POSITIVE SOLUTIONS OF SEMILINEAR HEAT EQUATIONS

2018 ◽  
Vol 100 (4) ◽  
pp. 291-308
Author(s):  
K. Achille Adou ◽  
K. Augustin Touré ◽  
A. Coulibaly
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
Numerical Study ◽  
Heat Equations
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