Non Diffusion Equation Research Articles

On the robustness of discontinuous patterns in degenerate reaction–diffusion systems with perturbed hysteresis

In this paper, we study the dynamics of a morphogenesis model which reproduces the formation of discontinuous patterns observed in numerous biological or ecological systems. This model is determined by a degenerate reaction–diffusion system with hysteresis in the non-diffusive equation. We focus on the robustness of discontinuous patterns under the action of a perturbation of the hysteresis process. We analyze the bifurcations of homogeneous stationary solutions to this nonlinear model and prove that the trivial solution is the only one to resist to a perturbation of strong intensity. We then prove an original result on the structural transformation of discontinuous patterns, which are seen to react by acquiring a supplementary discontinuity jump under the effect of a perturbation of small intensity. We interpret the supplementary jump for ecological systems as the possible emergence of an invasive ecosystem. Numerical simulations and animations are finally provided to guide intuition on the complex morphogenesis process of this dynamical system.

Exponential Runge-Kutta Parareal for non-diffusive equations

Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. This paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schrödinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations.

On the stability of exponential integrators for non-diffusive equations

Exponential integrators are a well-known class of time integration methods that have been the subject of many studies and developments in the past two decades. Surprisingly, there have been limited efforts to analyze their stability and efficiency on non-diffusive equations to date. In this paper we apply linear stability analysis to showcase the poor stability properties of exponential integrators on non-diffusive problems. We then propose a simple repartitioning approach that stabilizes the integrators and enables the efficient solution of stiff, non-diffusive equations. To validate the effectiveness of our approach, we perform several numerical experiments that compare partitioned exponential integrators to unmodified ones. We also compare repartitioning to the well-known approach of adding hyperviscosity to the equation right-hand-side. Overall, we find that the repartitioning restores convergence at large timesteps and, unlike hyperviscosity, it does not require the use of high-order spatial derivatives.

Correction to: Wellposedness and Convergence of Solutions to a Class of Forced Non-diffusive Equations with Applications

Correction to: Wellposedness and Convergence of Solutions to a Class of Forced Non-diffusive Equations with Applications

On the Non-diffusive Magneto-Geostrophic Equation

Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In Friedlander and Vicol (Nonlinearity 24(11)::3019–3042, 2011), the authors prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for periodic perturbations with initial data localized in frequency straight lines crossing the origin. For such well-prepared data, the local existence and uniqueness of solutions can be obtained in Sobolev spaces and the global existence holds under a size condition over the $$H^{5/2^{+}}({{\mathbb {T}}}^3)$$ norm of the perturbation.

Wellposedness and Convergence of Solutions to a Class of Forced Non-diffusive Equations with Applications

This paper considers a family of non-diffusive active scalar equations where a viscosity type parameter enters the equations via the constitutive law that relates the drift velocity with the scalar field. The resulting operator is smooth when the viscosity is present but singular when the viscosity is zero. We obtain Gevrey-class local well-posedness results and convergence of solutions as the viscosity vanishes. We apply our results to two examples that are derived from physical systems: firstly a model for magnetostrophic turbulence in the Earth's fluid core and secondly flow in a porous media with an "effective viscosity".

On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations

We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is globally well-posed (cf Friedlander and Vicol (2011 Ann. Inst. Henri Poincaré Anal. Non Linéaire 28 283–301)). In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.

Characterization of induction motors in adjustable-speed drives using a time-stepping coupled finite-element state-space method including experimental validation

This paper describes how a comprehensive time-stepping coupled finite element phase flux linkage-based state-space modeling approach can be used for the characterization of induction motors in adjustable-speed drives. The model implemented the faster indirect and iterative coupling approach based on the curl-curl nondiffusion equation rather than the direct coupling approach, which is based on the curl-curl diffusion equation. The model presented in this paper is capable of rigorously modeling the effects of magnetic nonlinearities and space harmonics due to the machine magnetic circuits' topology and winding layouts, time harmonics resulting from electronic switching of inverters, as well as the synergistic interaction between these time and space harmonics. This includes the presentation of a unique approach for the calculation of electromagnetic torque by means of a concept of energy balance computed from instantaneous magnetic field solutions calculated at each time sample in an at cycle. This approach reveals higher content of torque ripples than given by more conventional approaches to torque calculations. The results of motor drive performance simulations and corresponding test results are shown to correlate well here with current waveforms and torque averages. This includes sinusoidal excitation and six-switch inverter excitation with pulsewidth modulation switching pattern, respectively.

On the reactive and non-diffusive equations for zero mach number flow

In this paper it is established the well-posedness of the reactive and non-diffusive equations for zero Mach number flow. The solution is constructed by the method of successive approximations and convergence is established combining elliptic regularity estimates and energy estimates for hyperbolic equations. The solution of the linearized equations is built by a mixture of Galerkin and successive approximations methods.

Split Space-Marching Finite-Volume Method for Chemically Reacting Supersonic Flow

A space-marching finite-volume method employing a nonorthogonal coordinate system and using a split differencing scheme for calculating steady supersonic flow over aerodynamic shapes is presented. It is a second-order-accurate mixed explicit-implicit procedure that solves the inviscid adiabatic and nondiffusive equations for chemically reacting flow in integral conservation-law form. The relationship between the finite-volume and differential forms of the equations is examined and the relative merits of each discussed. The method admits initial Cauchy data situated on any arbitrary surface and integrates them forward along a general curvilinear coordinate, distorting and deforming the surface as it advances. The chemical kinetics term is split from the convective terms which are themselves dimensionally split, thereby freeing the fluid operators from the restricted step size imposed by the chemical reactions and increasing the computational efficiency. The accuracy of this splitting technique is analyzed, a sufficient stability criterion is established, a representative flow computation is discussed, and some comparisons are made with another method.