Quantitative Evaluation of Numerical Diffusion (Truncation Error)

1971 ◽  
Vol 11 (03) ◽  
pp. 315-320 ◽  
Author(s):  
R.B. Lantz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Truncation Error ◽  
Convective Diffusion ◽  
Block Size ◽  
Time Step ◽  
Partial Differential ◽  
Numerical Diffusion ◽  
Convection Diffusion

Abstract Numerical diffusion (truncation error) can limit the usefulness of numerical finite-difference approximations to solve partial differential equations. Many reservoir simulation users are aware of these limitations but are not as familiar with actually quantifying the magnitude of the truncation error. This paper illustrates that, over a wide range of block size and time step, the truncation error expressions for convective-diffusion partial differential equations are quantitative. Since miscible, thermal, and immiscible processes can be of the convective-diffusion equation form, the truncation error expressions presented can provide guidelines for choosing block size-time step combinations that minimize the effect of numerical diffusion. Introduction Truncation error limits the use of numerical finite-difference approximations to solve partial differential equations. In the solution of convection-diffusion equations, such as occur in miscible displacement and thermal transport, truncation error results in an artificial dispersion term often denoted as numerical diffusion. The differential equations describing two-phase fluid flow can also be rearranged into a convection-diffusion form. And, in fact, miscible and immiscible differential equations have been shown to be completely analogous. In this form, it is easy to infer that numerical diffusion will result in an additional term resembling flow due to capillarity. Many users of numerical programs, and probably all numerical analysts, recognize that the magnitude of the numerical diffusivity for convection-diffusion equations can depend on both block size and time step. Most expressions developed in the literature have been used primarily to determine the order of the error rather than to quantify it. The primary purpose of this paper is to give the user more than just a qualitative feel for the importance of truncation error. In this paper, insofar as possible, analytical expressions for truncation error are compared by experiment to computed values for the numerical diffusivity. Consequently, the reservoir simulator user can observe that these expressions are quantitative and can use them as guidelines for choosing block sizes and time steps that keep the numerical diffusivity small. DEVELOPMENT OF EXPRESSIONS FOR TRUNCATION ERROR APPLICATION TO CONVECTION-DIFFUSION EQUATION To illustrate the method of quantifying numerical diffusivity, consider a convective-diffusion equation of the form: ..............(1) Symbols are defined in the Nomenclature. The first term on the right-hand side represents the diffusion, and the second term represents convection. Such an equation describes the flow of either a two-component miscible mixture or heat in one dimension with constant diffusivity. EXPLICIT DIFFERENCE FORMS An explicit expression for the truncation error (the space derivatives are approximated at a known time level) can be developed by examining the Taylor's series expansion representing first- and second-order derivatives. For the time derivative: .....(2) SPEJ P. 315

Rigorous Calculation of Miscible Displacement Using Immiscible Reservoir Simulators

1970 ◽  
Vol 10 (02) ◽  
pp. 192-202 ◽  
Author(s):  
R.B. Lantz
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Capillary Pressure ◽  
Truncation Error ◽  
Capillary Flow ◽  
Miscible Displacement ◽  
Partial Differential ◽  
Two Phase ◽  
Numerical Diffusion ◽  
Three Phase

Abstract In the past miscible displacement calculations have been approximated with two-phase reservoir simulators. Such calculations have neglected diffusional mixing between miscible components. In fact, no analog bas been proposed for rigorously treating miscible simulations with two-phase programs. This paper describes requirements that programs. This paper describes requirements that permit such a rigorous simulation. permit such a rigorous simulation. The sets of partial differential equations describing each of the displacement processes are shown to be exactly analogous if relative permeability and capillary pressure functions are permeability and capillary pressure functions are adjusted in a special manner. Application of the "miscible" analogy in a two-phase simulator, however, has several limitations, the most severe of which is the truncation error (numerical diffusion) typical of an immiscible formulation. Since this error is time-step and/or block-size dependent, numerical smearing can, in principle, be made as small as necessary. But this feature limits the practical applicability of the "miscible" analogy practical applicability of the "miscible" analogy to cases with rather large physical diffusion. The range of applicability and other limitations are outlined in the paper. Also, illustrative sample calculations are presented for linear, radial and layer-cake systems. Component densities and viscosities are varied in the linear model. Introduction In recent years, use of two- and three-phase reservoir simulators to calculate the performance of immiscible fluid displacement has become widespread. Reservoir simulators capable of calculating miscible displacement problems, however, have been limited to special use programs. The primary reason for this limitation has been the significant truncation error (numerical diffusion) typical of ordinary finite difference approximations to the miscible equations. The method of characteristics provided a means of making miscible displacement calculations without significant truncation error. A recently proposed second calculation technique, based on variational methods, also reduces numerical diffusion. Both of these calculational techniques can be used for immiscible calculations. Still, general miscible displacement applications such as gas cycling, enriched-gas injection, or tracer injection have historically required use of immiscible reservoir simulators for performance predictions. Larson et al. have reported an example of such use of a two-phase computer program. Displacement involving two components flowing within a single phase would appear to be analogous to a two-phase displacement. Yet, past miscible calculations using immiscible simulators made the two-phase saturation profile as near piston-like as possible and neglected component mixing due to possible and neglected component mixing due to diffusional processes. The capillary pressure function was chosen to minimize capillary flow. Also, in these miscible approximations, no provision had been made for viscosity variations provision had been made for viscosity variations with component concentration. Though mixing due to diffusional processes had been neglected, countercurrent diffusion due to component concentration differences in a miscible process should be essentially analogous to countercurrent capillary flow due to saturation differences in a two-phase system. This paper describes a method by which two- and three-phase reservoir simulators can be made to calculate miscible displacement rigorously. The only requirement of the method is that relative permeability and capillary pressure be special permeability and capillary pressure be special functions of saturation. With these properly chosen functions, the set of partial differential equations describing immiscible displacement becomes completely analogous to the partial differential equations describing miscible displacement. SPEJ P. 192

FINITE ELEMENT METHODS FOR NONLINEAR ACOUSTICS IN FLUIDS

2007 ◽  
Vol 15 (03) ◽  
pp. 353-375 ◽  
Author(s):  
TIMOTHY WALSH ◽  
MONICA TORRES
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Acoustics ◽  
Time Step ◽  
Partial Differential ◽  
Time Step Size ◽  
Finite Element Discretizations ◽  
Acoustic Fluid ◽  
Element Discretizations

In this paper, weak formulations and finite element discretizations of the governing partial differential equations of three-dimensional nonlinear acoustics in absorbing fluids are presented. The fluid equations are considered in an Eulerian framework, rather than a displacement framework, since in the latter case the corresponding finite element formulations suffer from spurious modes and numerical instabilities. When taken with the governing partial differential equations of a solid body and the continuity conditions, a coupled formulation is derived. The change in solid/fluid interface conditions when going from a linear acoustic fluid to a nonlinear acoustic fluid is demonstrated. Finite element discretizations of the coupled problem are then derived, and verification examples are presented that demonstrate the correctness of the implementations. We demonstrate that the time step size necessary to resolve the wave decreases as steepening occurs. Finally, simulation results are presented on a resonating acoustic cavity, and a coupled elastic/acoustic system consisting of a fluid-filled spherical tank.

Numerical Solution of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method

2014 ◽  
Vol 6 (01) ◽  
pp. 107-119 ◽  
Author(s):  
D. B. Dhaigude ◽  
Gunvant A. Birajdar
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Discrete Schrödinger Equation ◽  
Adomian Decomposition

AbstractIn this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method. Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation, nonlinear fractional discrete Schrodinger equation, fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger’s equation. The obtained solution is verified by comparison with exact solution whenα= 1.

A hybrid analytical–numerical method for solving evolution partial differential equations. I. The half-line

2008 ◽  
Vol 464 (2095) ◽  
pp. 1823-1849 ◽  
Author(s):  
N Flyer ◽  
A.S Fokas
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Plane ◽  
Complex Analysis ◽  
Variable Coefficients ◽  
Integral Representations ◽  
New Method ◽  
Half Line ◽  
Time Step ◽  
Partial Differential

A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equations in cylindrical coordinates). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via the Fourier or Laplace transforms. Examples include the heat equation and the first and second versions of the Stokes equation for arbitrary initial and boundary data on the half-line. The new method has advantages in comparison with classical methods, such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries. As a result, these solutions are well suited for numerics, allowing the solution to be computed at any point in space and time without the need to time step. Simple deformation of the contours of integration followed by mapping the contours from the complex plane to the real line allow for fast and efficient numerical evaluation of the integrals.

A New Image Demising Method Based on Partial Differential Equations

2013 ◽  
Vol 443 ◽  
pp. 22-26
Author(s):  
Yong Xing Lin ◽  
Xiao Yan Xu ◽  
Xian Dong Zhang
Keyword(s):  
Image Processing ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Equation ◽  
Objective Function ◽  
Thermal Diffusion ◽  
Minimization Problem ◽  
Iterative Process ◽  
Function Minimization ◽  
Partial Differential

In the paper, we discuss the image demising models, based on partial differential equations. It is through the use of the concept of variations in the calculus of the objective function minimization problem, defines the image processing tasks. The results show that the model expands 2d thermal diffusion equation. Therefore, it is easy to get solution is to use a simple iterative process.

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