A comparison of conservative upwind difference schemes for the euler equations

Numerical results are presented and compared for three conservative upwind difference schemes for the Euler equations when applied to two standard test problems. This includes consideration of the effect of treating part of the flux balance as a source, and a comparison of different averaging of the flow variables. Two of the schemes are also shown to be equivalent in their implementation, while being different in construction and having different approximate Jacobians.     

A Galerkin procedure for systems of differential equations

A continuous time and an extrapolated coefficient Crank-Nicolson-Galerkin method are considered for approximating solutions of boundary and initial value problems for a quasi-linear parabolic system of partial differential equations which is coupled to a non-linear system of ordinary differential equations. A priori bounds are derived that reduce the estimation of error to problems in approximation theory. Then approximation theory results yield optimal order rates of convergence for theH ¹ (Ω) norm. The extrapolated coefficient method yields linear algebraic equations for strongly non-linear problems. this research was supported in part by the National Science Foundation Grant No. MCS 75-21317 and Energy-related Postdoctoral Fellowship at the University of Chicago.     

A novel immersed boundary procedure for flow and heat simulations with moving boundary

This article describes a novel immersed boundary procedure for computing the flow and heat transfer problems with moving and complex boundary. Although the immersed boundary techniques have been successfully implemented to these flow and heat simulations, a frequently encountered drawback of this method is the relatively low accuracy proximate to the boundary due to the spreading of forcing function or the interpolation scheme. In this study, we propose a moving-grid process under the arbitrary Lagrangian-Eulerian framework to reduce the numerical diffusion near the immersed boundary. The incompressible Navier-Stokes equations are discretized spatially using unstructured finite element method, and advanced temporally by an operator-splitting scheme. The methodology is validated by the simulations of flow induced by an oscillating cylinder in a free stream. The capability of the proposed method is further demonstrated by good predictions of flow passing the rotating fan in a channel and also flow driven by two independent rotating fans in a circular cavity.     

A new boundary layer matching procedure for singularly perturbed system

This short paper proposes a procedure for applying singular perturbation methods to separately analyze state dynamics even when they vary on the same time scale. The intent is to expand the family of problems to which these methods can be successfully applied. In particular, several problems in flight mechanics are identified. Numerical results for the minimum time to climb problem are given where the procedure is used to separate altitude and flight-path angle dynamics to produce a closed-form solution for lift.     

A robust solution procedure for hyperelastic solids with large boundary deformation

Compressible Mooney–Rivlin theory has been used to model hyperelastic solids, such as rubber and porous polymers, and more recently for the modeling of soft tissues for biomedical tissues, undergoing large elastic deformations. We propose a solution procedure for Lagrangian finite element discretization of a static nonlinear compressible Mooney–Rivlin hyperelastic solid. We consider the case in which the boundary condition is a large prescribed deformation, so that mesh tangling becomes an obstacle for straightforward algorithms. Our solution procedure involves a largely geometric procedure to untangle the mesh: solution of a sequence of linear systems to obtain initial guesses for interior nodal positions for which no element is inverted. After the mesh is untangled, we take Newton iterations to converge to a mechanical equilibrium. The Newton iterations are safeguarded by a line search similar to one used in optimization. Our computational results indicate that the algorithm is up to 70 times faster than a straightforward Newton continuation procedure and is also more robust (i.e., able to tolerate much larger deformations). For a few extremely large deformations, the deformed mesh could only be computed through the use of an expensive Newton continuation method while using a tight convergence tolerance and taking very small steps.     

A new method for solving boundary value problems for partial differential equations

This paper proposes a symmetry–iteration hybrid algorithm for solving boundary value problems for partial differential equations. First, the multi-parameter symmetry is used to reduce the problem studied to a simpler initial value problem for ordinary differential equations. Then the variational iteration method is employed to obtain its solution. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems.     

A meshless Galerkin method for Stokes problems using boundary integral equations

A meshless Galerkin scheme for the simulation of two-dimensional incompressible viscous fluid flows in primitive variables is described in this paper. This method combines a boundary integral formulation for the Stokes equation with the moving least-squares (MLS) approximations for construction of trial and test functions for Galerkin approximations. Unlike the domain-type method, this scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns, thus it is especially suitable for the exterior problems. Compared to other meshless methods such as the boundary node method and the element free Galerkin method, in which the MLS is also introduced, boundary conditions do not present any difficulty in using this meshless method. The convergence and error estimates of this approach are presented. Numerical examples are also given to show the efficiency of the method.     

Three-point boundary value problems for difference equations

In this paper, the existence and nonexistence of positive solutions for three-point nonlinear boundary value problems of difference equations are considered.     

Convergence in the boundary layer for singularly perturbed equations

A singularly perturbed linear finite-dimensional ordinary differential equation is considered on the half-line [0, ∞). The reduced system is assumed to have a unique solution in the sense of distributions. It is proved that under reasonable conditions the solution of the full system converges to that of the reduced in the distributional sense, and the order of the limiting distribution is computed. An application to the optimal linear-quadratic regulator with cheap control is described.     

A capillary free boundary problem governed by the Navier-Stokes equations

In this paper an unsteady capillary free boundary in a viscous imcompressible fluid is studied from the numerical point of view. The linearized problem is formulated, and existence and uniqueness of a solution is proved. The linearized problem is discretized by a finite element method in space and a finite difference method in time. The numerical scheme is proved to be convergent and stable. The method is applied to the computation of free boundaries, eigenfrequencies, damping factors and vibration modes. The influence of the initial conditions, the boundary conditions, the contact angle, the Ohnesorge number and the Bond number is studied.     

A new procedure for evaluating the time domain boundary influence matrix of unbounded media

The finite element method can only deal with finite domains with well defined boundaries. For dynamic problems involving unbounded media, the boundaries of the finite model distort the real physical behaviour of the problem if they remain untreated. For many problems it is possible to formulate so-called silent boundary conditions which perfectly simulate the effect of the truncated unbounded medium. Unfortunately, most of these conditions are properly formulated in the frequency domain.

The present paper introduces a new procedure which employs these frequency-dependent boundary conditions to calculate the time domain influence matrix of the truncated unbounded medium. This matrix, which was introduced in a previous publication, is used to calculate the reflection-free response of the truncation boundary one time step ahead of the present time station. The known boundary response is then used as a prescribed condition for the finite model. A one-dimensional example with a frequency-dependent boundary condition is presented to examine the effectiveness of the new procedure. Two other silent boundary conditions formulated directly in the time domain are also examined.     

A modified lobatto collocation for linear boundary value problems of differential-algebraic equations

The Lobatto collocation method is modified for efficiently solving linear boundary value problems of differential-algebraic equations with index 1. The stability and superconvergence of this method are established. Numerical implementations are discussed and a numerical example is given.     

A method of iteration for boundary value solution of ordinary differential equations

A globally stable iterative search algorithm for finding the unknown initial values in the two-point boundary-value problem (TPBVP) arising in ordinary differential equations has been developed. The problem can be embedded in a discrete-time feedback control system, the steady state of which is the required solution. Algorithms are found through the construction of globally stable digital controllers.     

Constraint preserving boundary conditions for the Ideal Newtonian MHD equations

We study and develop constraint preserving boundary conditions for the Newtonian magnetohydrodynamic equations and analyze the behavior of the numerical solution upon considering different possible options. We concentrate on both the standard ideal MHD system and the one augmented by a “pseudo potential” to control the divergence free constraint. We show how the boundary conditions developed significantly reduce the violations generated at the boundaries at the numerical level and how lessen their influence in the interior of the computational domain by making use of the available freedom in the equations.     

Three-point boundary value problems for second-order discrete equations

We formulate existence results for solutions to discrete equations which approximate three-point boundary value problems for second-order ordinary differential equations.     

Periodic boundary value problems for neutral multi-pantograph equations

In this paper, a new method is proposed for solving nonlinear neutral multi-pantograph equations with periodic boundary conditions by combining the advantages of the homotopy perturbed method (HPM) and the reproducing kernel method (RKM). The numerical results obtained show that the present method is reliable.     

A three-point boundary value problem with an integral two-space-variables condition for parabolic equations

In this paper, we study a three-point boundary value problem with an integral two-space-variables condition for a class of parabolic equations. The existence and uniqueness of the solution in the functional weighted Sobolev space are proved. The proof is based on two-sided a priori estimates and on the density of the range of the operator generated by the problem considered.     

A one sweep method for linear difference equations with two point boundary conditions

A new method is proposed for reducing two point boundary value problems for systems of linear difference equations to initial value problems. The method has the advautage that only one sweep is required. Comparisons against classical reduction methods are made. Supported by the National Institutes of Health under Grants No. GM-16197-01 and GM-16437-01.     

A collocation method for boundary value problems of differential equations with functional arguments

A collocation procedure with polynomial and piecewise polynomial approximation is considered for second order functional differential equations with two side-conditions. The piecewise polynomials are taken in the classC ¹ and reduce to polynomials of increasing degree on each interval of a suitable assigned partition. Appropriate choices of the partition are made, according to the jump discontinuities in the derivatives caused by the functional argument, in order to optimize the rate of convergence.