Unsteady Partial Differential Equations Research Articles

A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media

In this paper we derive a posteriori error estimates for the compositional model of multiphase Darcy flow in porous media, consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations. We show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators. We then decompose the estimators into the space, time, linearization, and algebraic error components. This allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error. Moreover, the spatial and temporal error components can be balanced by time step and space mesh adaptation. Our analysis applies to a broad class of standard numerical methods, and is independent of the linearization and of the iterative algebraic solvers employed. We exemplify it for the two-point finite volume method with fully implicit Euler time stepping, the Newton linearization, and the GMRes algebraic solver. Numerical results on two real-life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria, already on fixed meshes, without any noticeable loss of precision.

Improvement of Stability and Accuracy of Time-Evolution Equation by Implicit Integration

The stability and accuracy of the numerical integration of the time-evolution equation obtained by discretizing an unsteady partial differential equation with respect to space variables has a crucial importance in solving the unsteady partial differebtial equation numerically. The second and fourth Runge-Kutta mrthods are widely used in the numerical calculation. However, in some cases, the stability is not sufficient. New implicit methods are proposed to increase stability and accuracy of the solution of the time-evolution-equation. Three new implicit methods, that is, implicit method usng linear approximation (IMP1), one using parabolic approximation (IMP2) and one using cubic approximation (IMP3) are proposed. In the case of linear problem, IMP1 is identical to the implicit method by Crank and Nicholson. The stability of various methods including Runge-Kutta method is discussed theoretically and numerically, and the numerical examples are shown to show the effectiveness of the implicit methods. It is proposed that the most practical way to increase both the accuracy and the stability in the solution of boundary value problems may be to use IMP1 and the smaller spatial mesh size.

Splitting schemes for poroelasticity and thermoelasticity problems

In this work, we consider the coupled systems of linear unsteady partial differential equations, which arise in the modelling of poroelasticity processes. Stability estimates of weighted difference schemes for the coupled system of equations are presented. Approximation in space is based on the finite element method. We construct splitting schemes and give some numerical comparisons for typical poroelasticity problems. The results of numerical simulation of a 3D problem are presented. Special attention is given to using high performance computing systems.

Modelling and simulation methodology of channel cooling using hydrocarbon fuel as coolant under supercritical pressures

A one-dimensional cooling channel model is proposed using unsteady partial differential equations (PDEs) and taking into account the strong dependencies of hydrocarbon fuel properties on temperature and pressure. The model can be used to evaluate the performance of channel cooling using hydrocarbon fuel as coolant under supercritical pressures, and to guide the design of a thermal protection system for aerospace application. Meanwhile, a graphical simulation method is developed to directly solve the proposed model using Simulink by converting the PDE to a set of ordinary differential equations without any need to write program codes. The graphical method can reduce the substantive work resulting from program codes writing and debugging and bring about interactions with users through its graphical user interface. In order to show the usability of the proposed model and the developed graphical method, n-pentane, a typical hydrocarbon fuel, is taken as an example to run two simulations for steady and transient calculations. The developed graphical simulator for channel cooling is verified by comparing the calculation results with the experimental data of an electrical heating tube. It is believed that this investigation will pave the way for implementation of the graphical simulation of the whole cooled propulsion system used for supersonic or hypersonic flights.

Two-layer film flow on a rotating disk: A numerical study

Development of thin two-layer film over a uniformly rotating disk is studied numerically under the assumption of planar interface and free surface. Similarity transformation is applied to transform the Navier–Stokes equations into a set of coupled non-linear, unsteady partial differential equations. This set of equations are solved numerically by using the finite-difference technique. It is observed that the rate of film thickness varies at different time zone depending on the rate of rotational speed of the disk. A physical explanation is provided to justify this anomalous behaviour. It is observed that, smaller thickness on the top layer enhance the initial rate of film thinning. But the overall effect of density, viscosity and the initial film thickness ratio are found to be insensitive to the final film thickness at large time.

Integrated gas dynamic computational modelling and thermodynamic combustion diagnostics of multicylinder four-stroke spark ignition engine using compressed natural gas as a fuel

A comprehensive computational simulation model has been developed to describe the performance, efficiency and emission characteristics of the four-stroke multi-cylinder spark ignition engine which uses compressed natural gas as a fuel. This model performs an integrated simulation of thermodynamic, gas dynamic, and chemical kinetics of the whole engine system coupling with intake and exhaust manifolds. The thermodynamic combustion process is modelled by using the turbulent flame propagation process with a two-zone chamber (burned and unburned). Gas dynamics in the manifold system are modelled by using unsteady compressible hyperbolic partial differential equations. These equations contain heat transfer, friction, area change in pipe and empirical constants, which are used to determine boundary conditions at pipe junctions, valves and open ends. These equations are transferred into a set of ordinary differential equations by applying the method of characteristics and are solved by the finite difference method. Regarding exhaust emissions, nitric oxide concentrations have been predicted by using the rate kinetic model in the power cycle and along the exhaust pipes. Carbon monoxide is computed under the chemical equilibrium condition and then the empirical adjustment is made for kinetic behaviours based upon experimental results. Finally, predicted results of this model have been compared with experimental results and are found to be compatible. The parametric studies have been conducted with the equivalence ratio and the engine RPM, to predict engine performance. Also, the variations of pressure, temperature, and gas compositions in the exhaust system have been studied.

Explicit Analytical Solutions of Linear and Nonlinear Interior Heat and Mass Transfer Equation Sets for Drying Process

Some algebraic explicit analytical solutions of the unsteady one-dimensional variable coefficient partial differential equation set describing drying process of an infinite plate are derived and given. Such explicit solutions have not yet been published before. Besides their irreplaceable theoretical value, analytical solutions can also serve as benchmark to check the results of recently rapidly developing numerical calculation and study various computational methods. In addition, some mathematical skills used in this paper are special and deserve further attention.

SPRINT2D

SPRINT2D is a set of software tools for solving both steady an unsteady partial differential equations in two-space variables. The software consists of a set of coupled modules for mesh generation, spatial discretization, time integration, nonlinear equations, linear algebra, spatial adaptivity, and visualization. The software uses unstructured triangular meshes and adaptive local error control in both space and time. The class of problems solved includes systems of parabolic, elliptic, and hyperbolic equations; for the latter by use of Riemann-solver-based methods. This article describes the software and shows how the adaptive techniques may be used to increase the reliability of the solution for a Burgers' equations problem, an electrostatics problem from elastohydrodynamic lubrication, and a challenging gas jet problem.

Numerical simulation for capillary electrophoresis: I. Development of a simulation program with high numerical stability

A computer program was developed for the simulations of capillary electrophoresis on the basis of unsteady partial differential equations. Since a new algorithm (shifted upwinding approximation) was used to suppress numerical divergence and oscillation, the program succeeded in generating only physically meaningful peaks at the cost of slight overestimation of ionic diffusion. The mathematical model and equations used in the program were described in detail, and simulation performance was compared among the previously reported programs, confirming the numerical stability of our program. As an application example, a system peak in indirect UV detection was simulated.

Modelling of stochastic heat transfer in a solid

The unsteady partial differential equations for expectation and correlation distributions of the stochastic temperature distribution in a solid are obtained, when the coefficients and the source term in the stochastic heat transfer equations are white Gaussian processes. Some solutions of the unsteady partial differential equations for expectation and correlation distributions of stochastic heat transfer are presented.

APPLICATION OF THE PIECEWISE PARABOLIC FINITE ANALYTIC METHOD TO THE THREE-DIMENSIONAL CAVITY FLOW

Abstract The three-dimensional piecewise parabolic finite analytic method (PPFAM3D), with vector potential and vorticity as dependent variables, was used to solve the steady three-dimensional (3-D) laminar cavity flow. Predictions were obtained for Reynolds numbers between 100 and 2000. Results for Re = 100, 400, and 1000 are compared with those available in the literature. The overall agreement is excellent. Results are also compared with the two-dimensional PPFAM cavity flow, showing the effect of three-dimensionality on the velocity predictions at the symmetry plane. The PPFAM3D is a robust numerical algorithm requiring no relaxation to produce physically meaningful and numerically converged solutions. It is a promising tool for solving a variety of flow phenomena governed by unsteady second-order partial differential convection-diffusion equations

Parametric studies on heat and mass transfer in metal hydride beds

The mathematical modelling of metal hydride beds of rectangular configuration has been carried out. The coupled heat and hydrogen transfer that takes place during absorption/desorption processes has been described by non-linear, unsteady partial differential equations. All parameters have been non-dimensionalised, thereby making the analysis a generalised one. The important alloy properties and operating conditions have been varied over a large range to make the results applicable to a variety of alloys and operating conditions. The effect of the important non-dimensional parameters on heat and mass transfer is studied. The results show the importance of heat-transfer enhancement in view of the very low thermal conductivity of hydride beds and fast kinetics.

A time-parallel multigrid-extrapolation method for parabolic partial differential equations

We consider the problem of solving unsteady partial differential equations on an MIMD machine. Conventional parallel methods use a data partitioning type approach in which the solution grid at each time-step is divided amongst the available processors. The sequential nature of the time integration is, however, retained. The algorithm presented in this paper makes use of a time-parallel approach, whreby several processors may be employed to solve at several time-steps simultaneously. The time-parallel method enables the inherent parallelism of the extrapolation scheme to be efficiently exploited, allowing a significant increase both in accuracy and in the degree of parallelism. The efficiencies obtained by an implementation on a message-passing multiprocessor demonstrate the suitability of the time-parallel extrapolation method for this type of equation.

TRANSIENT FREE CONVECTION ABOUT A VERTICAL FLAT PLATE EMBEDDED IN A SATURATED POROUS MEDIUM

An analysis is presented of the transient, buoyancy-induced flow and heat transfer in a Darcian fluid-saturated porous medium adjacent to a suddenly heated semi-infinite vertical wall. Transient profiles of the temperature field and the local Nusselt number are obtained by solving the unsteady boundary layer partial differential equations numerically. It is shown that the governing equations are of a singular parabolic type and can be solved accurately in a semi-similar, finite elliptic domain using a successive relaxation method. The solution thus obtained is compared with a solution obtained in the physical domain using an explicit upwind method. The results confirm that during the initial stage, before the effects of the leading edge are influential at a location, heat transfer and flow phenomena in porous media are governed by transient one-dimensional conduction. New data are presented for the transition from this initial conduction stage to a fully two-dimensional transient, which ultimately termina...

Anisotropic waves in elastoplastic soils

The Prandtl-Reuss theory extended to consider a medium which is compressible in the plastic region is utilized to study waves in elastoplastic soils. The von Mises yield condition is applied with the yield stress a function of the pressure, which is typical of soils. An arbitrary pressure-density relationship with noncoincident loading and unloading paths is also allowed. The full nonlinear three-dimensional unsteady partial differential equations are analyzed by determining their characteristic or discontinuity surfaces. The characteristic analysis results in equations for the wave speed of infinitesimal disturbances in the medium as a function of the orientation of the wave normal vector, the stress, the shear modulus, the bulk modulus and the density. These equations have been coded for digital computer solution and automatic plotting of the three-dimensional normal wave speed diagrams and their corresponding wave surfaces or fronts. Results in the form of normal wave speed diagrams and wave surfaces reveal the nature of the anisotropic waves which are found in soils in the nonlinear elastic and plastic regimes. The variation of the waves with variation in the parameters is also found. Most of the phenomena of cusped wave surfaces and intersecting normal wave speed diagrams found in anisotropic elastic waves, magnetofluid dynamics and crystal optics are also observed in the present results. In addition, it is found that certain waves do not propagate (their wave speeds become imaginary) in directions normal to the axis of the principal deviatoric stress whose absolute value is maximum.