Multi-stage numerical method of collocations for solving second-order ODEs

Рассматривается задача цифрового моделирования выходного массопотока смесителя непрерывного действия сыпучих материалов, используемого в пищевой промышленности. Для построения модели используются прямое и обратное преобразования Эйлера, билинейное преобразование, а также непрерывные дроби. Сравнительный анализ показал, что использование авторского подхода на основе непрерывных дробей приводит к наиболее адекватным цифровым моделям массопотоков смесителя непрерывного действия

Confinement of rotating convection by a laterally varying magnetic field

Spherical shell dynamo models based on rotating convection show that the flow within the tangent cylinder is dominated by an off-axis plume that extends from the inner core boundary to high latitudes and drifts westward. Earlier studies explained the formation of such a plume in terms of the effect of a uniform axial magnetic field that significantly increases the lengthscale of convection in a rotating plane layer. However, rapidly rotating dynamo simulations show that the magnetic field within the tangent cylinder has severe lateral inhomogeneities that may influence the onset of an isolated plume. Increasing the rotation rate in our dynamo simulations (by decreasing the Ekman number $E$) produces progressively thinner plumes that appear to seek out the location where the field is strongest. Motivated by this result, we examine the linear onset of convection in a rapidly rotating fluid layer subject to a laterally varying axial magnetic field. A cartesian geometry is chosen where the finite dimensions $(x,z)$ mimic $(\phi,z)$ in cylindrical coordinates. The lateral inhomogeneity of the field gives rise to a unique mode of instability where convection is entirely confined to the peak-field region. The localization of the flow by the magnetic field occurs even when the field strength (measured by the Elsasser number $\varLambda$) is small and viscosity controls the smallest lengthscale of convection. The lowest Rayleigh number at which an isolated plume appears within the tangent cylinder in spherical shell dynamo simulations agrees closely with the viscous-mode Rayleigh number in the plane layer linear magnetoconvection model. The localized excitation of viscous-mode convection by a laterally varying magnetic field provides a mechanism for the formation of isolated plumes within Earth's tangent cylinder.Comment: 12 figures, 3 table

A numerical comparison of Chebyshev methods for solving fourth order semilinear initial boundary value problems

In solving semilinear initial boundary value problems with prescribed non-periodic boundary conditions using implicit–explicit and implicit time stepping schemes, both the function and derivatives of the function may need to be computed accurately at each time step. To determine the best Chebyshev collocation method to do this, the accuracy of the real space Chebyshev differentiation, spectral space preconditioned Chebyshev tau, real space Chebyshev integration and spectral space Chebyshev integration methods are compared in the L2 and W2,2 norms when solving linear fourth order boundary value problems; and in the L∞([0,T]; L2) and L∞([0,T]; W2,2) norms when solving initial boundary value problems. We find that the best Chebyshev method to use for high resolution computations of solutions to initial boundary value problems is the spectral space Chebyshev integration method which uses sparse matrix operations and has a computational cost comparable to Fourier spectral discretization.</p