1,726 research outputs found
Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms
Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear time-delayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters , and the resulting equations together with the two-point boundary conditions constitute a system of ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms
Structure and stability of non-symmetric Burgers vortices
We investigate, numerically and analytically, the structure and stability of steady and quasi-steady solutions of the Navier–Stokes equations corresponding to stretched vortices embedded in a uniform non-symmetric straining field, ([alpha]x, [beta]y, [gamma]z), [alpha]+[beta]+[gamma]=0, one principal axis of extensional strain of which is aligned with the vorticity. These are known as non-symmetric Burgers vortices (Robinson & Saffman 1984). We consider vortex Reynolds numbers R=[Gamma]/(2[pi]v) where [Gamma] is the vortex circulation and v the kinematic viscosity, in the range R=1[minus sign]104, and a broad range of strain ratios [lambda]=([beta][minus sign][alpha])/([beta]+[alpha]) including [lambda]>1, and in some cases [lambda][dbl greater-than sign]1. A pseudo-spectral method is used to obtain numerical solutions corresponding to steady and quasi-steady vortex states over our whole (R, [lambda]) parameter space including [lambda] where arguments proposed by Moffatt, Kida & Ohkitani (1994) demonstrate the non-existence of strictly steady solutions. When [lambda][dbl greater-than sign]1, R[dbl greater-than sign]1 and [epsilon][identical with][lambda]/R[double less-than sign]1, we find an accurate asymptotic form for the vorticity in a region 11. An iterative technique based on the power method is used to estimate the largest eigenvalues for the non-symmetric case [lambda]>0. Stability is found for 0[less-than-or-eq, slant][lambda][less-than-or-eq, slant]1, and a neutrally convective mode of instability is found and analysed for [lambda]>1. Our general conclusion is that the generalized non-symmetric Burgers vortex is unconditionally stable to two-dimensional disturbances for all R, 0[less-than-or-eq, slant][lambda][less-than-or-eq, slant]1, and that when [lambda]>1, the vortex will decay only through exponentially slow leakage of vorticity, indicating extreme robustness in this case
Nonlinear transient waves in coupled phase oscillators with inertia
Like the inertia of a physical body describes its tendency to resist changes
of its state of motion, inertia of an oscillator describes its tendency to
resist changes of its frequency. Here we show that finite inertia of individual
oscillators enables nonlinear phase waves in spatially extended coupled
systems. Using a discrete model of coupled phase oscillators with inertia, we
investigate these wave phenomena numerically, complemented by a continuum
approximation that permits the analytical description of the key features of
wave propagation in the long-wavelength limit. The ability to exhibit traveling
waves is a generic feature of systems with finite inertia and is independent of
the details of the coupling function.Comment: 12 pages, 4 figure
Structure and stability of non-symmetric Burgers vortices
We investigate, numerically and analytically, the structure and stability of steady and quasi-steady solutions of the Navier–Stokes equations corresponding to stretched vortices embedded in a uniform non-symmetric straining field, ([alpha]x, [beta]y, [gamma]z), [alpha]+[beta]+[gamma]=0, one principal axis of extensional strain of which is aligned with the vorticity. These are known as non-symmetric Burgers vortices (Robinson & Saffman 1984). We consider vortex Reynolds numbers R=[Gamma]/(2[pi]v) where [Gamma] is the vortex circulation and v the kinematic viscosity, in the range R=1[minus sign]104, and a broad range of strain ratios [lambda]=([beta][minus sign][alpha])/([beta]+[alpha]) including [lambda]>1, and in some cases [lambda][dbl greater-than sign]1. A pseudo-spectral method is used to obtain numerical solutions corresponding to steady and quasi-steady vortex states over our whole (R, [lambda]) parameter space including [lambda] where arguments proposed by Moffatt, Kida & Ohkitani (1994) demonstrate the non-existence of strictly steady solutions. When [lambda][dbl greater-than sign]1, R[dbl greater-than sign]1 and [epsilon][identical with][lambda]/R[double less-than sign]1, we find an accurate asymptotic form for the vorticity in a region 11. An iterative technique based on the power method is used to estimate the largest eigenvalues for the non-symmetric case [lambda]>0. Stability is found for 0[less-than-or-eq, slant][lambda][less-than-or-eq, slant]1, and a neutrally convective mode of instability is found and analysed for [lambda]>1. Our general conclusion is that the generalized non-symmetric Burgers vortex is unconditionally stable to two-dimensional disturbances for all R, 0[less-than-or-eq, slant][lambda][less-than-or-eq, slant]1, and that when [lambda]>1, the vortex will decay only through exponentially slow leakage of vorticity, indicating extreme robustness in this case
Stability of equilibrium solutions of a double power reaction diffusion equation with a Dirac interaction
In this paper we provide detailed information about the instability of
equilibrium solutions of a nonlinear family of localized reaction-difussion
equations in dimensione one. Beyond we provide explicit formulas to the
equilibrium solutions, via perturbation method and we calculate the exact
number of positive eigenvalues of the linear operator associated to the
stability problem, which allow us to compute the dimension of the unstable
manifold.Comment: 16 pages, 3 figure
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