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Abstract We show that local weak solutions to anisotropic elliptic equations with bounded and measurable coefficients, whose prototype is are continuous. Our analysis exploits a particular intrinsic geometry that, within the limiting case on the exponents, allows for a reduction of the oscillation of the solution.
A method is presented for the solution of the full potential equation in conservation form. This assures that a weak solution satisfying proper isentropic jump conditions is obtained, giving an improved representation of shock waves in comparison with earlier nonconservative schemes. The method uses the concept of artificial viscosity to produce a stable difference scheme in the supersonic zone, and artificial time to generate a convergent iterative scheme for the solution of the difference equations. Results of calculations using the nonconservative and conservative schemes are compared.
Consideration is given to the numerical simulation of the solution growth of triglycine sulfate in space, with heat transfer, mass transfer, buoyancy-driven convection, the dependence of solubility on temperature, and finite interface kinetics taken into account. In the bulk solution, weak thermal convection was established quickly. The thermal convection then becomes dominated by solutionally driven motion. The convection due to the steady background g is predicted to influence both the growth rate and the crystal morphology. The time required for buoyancy-driven convection to influence the growth rate depends on the magnitude of g, i.e., more time for smaller g. Under the same cooling condition, the average growth rate increases as the magnitude of g increases. A steady average growth rate of 1.0 mm/d can be obtained when a properly designed cooling rate is applied to the sting. The most uniform growth is obtained when the g vector is aligned such that the solution flows normal towards the center of the crystal surface.
We present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier–Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.
The nucleation and growth of different phases of simulated polar stratospheric cloud (PSC) particles were investigated in the laboratory. Solutions and mixtures of solutions at concentrations 1 to 5 m (molality) of ammonium sulfate, ammonium bisulfate, sodium chloride, sulfuric acid, and nitric acid were supercooled to prescribed temperatures below their equilibrium melting point. These solutions were contained in small diameter glass tubing of volumes ranging from 2.6 to 0.04 ml. Samples were nucleated by insertion of an ice crystal, or in some cases by a liquid nitrogen cooled wire. Crystallization velocities were determined by timing the crystal growth front passages along the glass tubing. Solution mixtures containing aircraft exhaust (soot) were also examined. Crystallization rates increased as deltaT2, where deltaT is the supercooling for weak solutions (2 m or less). The higher concentrated solutions (greater than 3 m) showed rates significantly less than deltaT2. This reduced rate suggested an onset of a glass phase. Results were applied to the nucleation of highly concentrated solutions at various stages of polar stratospheric cloud development within the polar stratosphere.
An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation-law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.
A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. In this work, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.
A new method of calculating transonic flows based on a 'rotated' difference scheme is described. It is suitable for the calculation of both two- and three-dimensional flows without restriction on the speed at infinity and is well adapted to computer use. The Murman procedure is modified to eliminate any assumptions about the direction of flow when constructing the difference scheme. The proper directional property is obtained by rotating the difference scheme to conform with the local stream direction. In the hyperbolic region retarded difference formulas are used for all contributions to the streamwise second derivative, producing a correctly oriented positive artificial viscosity. In the absence of a simple implicit scheme in the hyperbolic and elliptic regions, the concept of iterations as steps in artificial time is introduced. Computer testing of this procedure provides numerical confirmation of the existence and uniqueness of weak solutions of the potential equation when a suitable entropy inequality is enforced.
Stability of stochastic processes defined by difference differential equations, noting properties of solution, weak infinitesimal operator, etc
Softening blueberry skin by treatment with weak solution of sodium hydroxide prior to freeze dehydration prevents tough, chewy skins upon rehydration.
An efficient numerical method for calculating plane, axisymmetric, and fully three-dimensional blunt-body flow is presented. It is a second-order-accurate, time-dependent finite-volume procedure that solves the Euler equations in integral conservation-law form. These equations are written with respect to a Cartesian coordinate system in which an embedded mesh adjusts in time to the motion of the bow shock that is automatically captured as part of the weak solution. With such an adjusting mesh, oscillations in flow properties near the shock are shown to be virtually eliminated. The scheme uses a time-splitting concept that accelerates the convergence appreciably. Comparisons are made between computed and experimental results.
A finite element algorithm for solving unsteady, three-dimensional high velocity impact problems is presented. A computer program was developed based on the Eulerian hydroelasto-viscoplastic formulation and the utilization of the theorem of weak solutions. The equations solved consist of conservation of mass, momentum, and energy, equation of state, and appropriate constitutive equations. The solution technique is a time-dependent finite element analysis utilizing three-dimensional isoparametric elements, in conjunction with a generalized two-step time integration scheme. The developed code was demonstrated by solving one-dimensional as well as three-dimensional impact problems for both the inviscid hydrodynamic model and the hydroelasto-viscoplastic model.
Extensions of a second order high resolution explicit method for the numerical computation of weak solutions of one dimensonal hyperbolic conservation laws are discussed. The main objectives were (1) to examine the shock resoluton of Harten's method for a two dimensional shock reflection problem, (2) to study the use of a high resolution scheme as a post-processor to an approximate steady state solution, and (3) to construct an implicit in the delta-form using Harten's scheme for the explicit operator and a simplified iteration matrix for the implicit operator.
A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurae scheme to an appropriately modified flux function. The so derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme.
The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme.
Consider a scalar, nonlinear conservative difference scheme satisfying the entropy condition. It is shown that difference schemes containing more numerical viscosity will necessarily converge to the unique, physically relevant weak solution of the approximated conservation equation. In particular, entropy satisfying convergence follows for E schemes - those containing more numerical viscosity than Godunov's scheme.
The present investigation has the objective to review the general structure of system of conservation laws with entropy, giving particular attention to a presentation of symmetric formulations of the equations of gas dynamics. Systems of hyperbolic conservation laws are considered, taking into account the case in which a system of equations is symmetric hyperbolic, questions of symmetrization, and entropy functions. The Euler equations for a polytropic gas in conservation form are considered. The symmetric form retains the conservation properties of the equations. Weak solutions remain, therefore, unchanged. Of particular computational interest is the possibility of using the symmetric form.
A class of new explicit second order accurate finite difference schemes for the computation of weak solutions of hyperbolic conservation laws is presented. These highly nonlinear schemes are obtained by applying a nonoscillatory first order accurate scheme to an appropriately modified flux function. The so-derived second order accurate schemes achieve high resolution while preserving the robustness of the original nonoscillatory first order accurate scheme. Numerical experiments are presented to demonstrate the performance of these new schemes.