Journal ArticleDOI

# An analysis of finite-difference and finite-volume formulations of conservation laws

01 Mar 1989-Journal of Computational Physics (Academic Press Professional, Inc.PUB19San Diego, CA, USA)-Vol. 81, Iss: 1, pp 1-52
TL;DR: In this article, a coordinate-free formulation of conservation laws is developed, which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system, and the analysis considers general types of equations: potential, Euler, and Navier-Stokes.

### INTRODUCTION

• Many current algorithms in computational fluid dynamics are based on the numerical solution of conservation laws.
• This choice is motivated by several considerations, the chief one being the ability to treat flow discontinuities automatically.
• There appears to be some confusion and ignorance concerning the relation of these two approaches.
• This necessitates the introduction of a novel treatment of flux vectors and flux Jacobian matrices solely in terms of physical vectors, and independent of a coordinate system.
• The next section presents the discretization of the equations, beginning with a careful discussion of grid definition.

### FORMULATION OF CONSERVATION LAWS

• Since physical conservation only has meaning over a finite region of space and a finite interval of time, the authors divide the flow region into contiguous cells which can vary with time.
• Let v(t) be the fluid velocity and surface element velocity, respectively.
• For nonequilibriumflow, the partial density of various species and the energy of internal states of species can be additional variables.
• This case is not discussed in this paper.).
• There are additional global conservation laws that can be derived under special assumptions that are sometimes used as constraints in an algorithm.

### S f

• The conservation of volume for a time-varying cell is given by An additional geometric constraint is that the sum of the cell volumes must equal the total volume of the flow region.
• A main motivation for the conservative formulation is to capture flow discontinuities in inviscid flow.
• The jump conditions across such discontinuities are just limiting forms of the integral relations.

### J

• Actually, since the geometry of space is continuous, Eq. ( 10) can be replaced by the weaker relation EQUATION.
• It is possible for algorithms to satisfy the weak form of the jump conditions and violate strict conservation.

### NEW COORDINATE-FREE TREATMENT OF FLUX VECTORS

• The authors have seen that Eq. (1) can represent a scalar or vector conservation law.
• The related higher order quantity is a tensor.
• Since all calculations must ultimately be done with numbers, and to avoid the confusion between the two uses of the term vector, the momentum conservation law is normally treated as three scalar laws for the components of the momentum.
• A compactness and greater physical insight can be obtained by retaining the physical vectors as components of the algebraic vectors.
• The procedure will be illustrated for the inviscid flow of a perfect gas.

### Vector Formulation of Flux Jacobian Matrices.

• This can be rewritten in the form of a matrix multiplication as where the dot product is implied in multiplying the second element of the row oector by the second element of the column oector.
• For some algorithms, one requires the diagonalization of A in the form R-'AR = A, where A is the eigenvalue matrix, and R and R--' are the right and left eigenvector matrices, respectively.
• For multiple space dimensions one requires a set of linearly independent eigenvectors corresponding to a repeated eigenvalue.
• For three dimensions, A is a five by five matrix, while R has five columns and R-' has five rows.
• One can define functions of the matrix A through where the projection operators PI, P2, and P3 are matrices satisfying.

### 1 -b3 b2u

• They must therefore be used with some care.
• As an example, the product R-'R gives the identity matrix.
• On the other hand, the product RR-' is undefined, and does not give the unit matrix I , even when interpreted as a tensor product.

### R o e Averaging.

• Similarly, the third component of Eq. (38) then yields.
• The sound speed, derived from the total enthalpy via c2 = rc(Hfu u), can be written as The above derivation is much more direct than that found in Ref.
• It follows from Eq. (30) that for either A 2 or AB, it is possible that the Roe averaged eigenvalue could lie outside the range determined by the two states L and R .
• The implications for algorithms based on Roe's scheme should be studied further.

### Flux Vector Splitting.

• The earliest methods utilize the homogeneity property (It is thus valid for a gas that is thermally perfect, but calorically imperfect.).
• A secondary grid can be obtained by determining the centers or centroids of the primary cells (in a non-unique way) and connecting them across cell faces.
• Standard finite-difference methods are based on the discretization of Eq. (20) , with the geometric quantities included in the definitions of the transformed variables.
• First of all, the geometric quantities must be evaluated using one-sided differences.
• The boundary procedure used to evaluate the flow variables at a boundary grid point may not explicitly satisfy Eq. (20) , which is used at the neighboring interior point.

### F init e-Volume Geometry .

• As a building block for subsequent calculations, the authors consider a triangular face whose vertices are rl, r2 and r3.
• Thus Eq. (55) is valid for any face with straight line edges.
• The above formulas can be used to make geometric calculations for an arbitrary cell with straight line edges.
• Each polygonal face can be subdivided into plane triangular facets, and the total volume treated as a sum of tetrahedra.
• The form (60b) is in terms of the two vectors joining opposite edge midpoints.

### The earliest expression for the volume of a hexahedron, based on an equivalent

• Plane face containing a diagonal, was given by Rizzi (Ref. 9).
• Kordulla and Vinokur (Ref. 10) showed that of the eight consistent divisions of the faces by diagonals, four result in a very simple expression for the volume.
• If one vertex of a cell main diagonal is chosen as the common apex, and the other vertex as the intersection of three equivalent plane faces, the six pyramids reduce to three pyramids sharing the main diagonal as a common edge.
• It was shown by Davies and Salmond (Ref. 12) that the same equivalent plane corresponds to a face defined as a doubly-ruled surface.

### Finite-Volume Discretization.

• Finite-volume methods are based on the discretization of Eq. (l), with the surface integral replaced by tEc sum of integrals over the faces of the cell.
• The method is normally applied to cells defined by the primary grid, so that certain cell faces will coincide with the flow region boundary.
• This insures that all the geometric identities and constraints are precisely satisfied.
• If F is spatially uniform (which is valid for a uniform free stream), the numerical calculation of the surface integral for each face should sum to zero (within roundoff errors).
• Thus the geometry of the discretization is treated separately from the treatment of the physical variables.

### 1 vi,j,k

• This will result in oscillations for a uniform flow since Eq. ( 6) is not This takes no more operations than the central difference formula and eliminates errors for a uniform flow.
• For problems in which the free stream is a uniform flow, an alternate procedure suggested in Ref. 17 is to subtract the free-stream fluxes from the conservation equations.
• This will guarantee exact cancellation of free-stream.
• To circumvent the need for artificial smoothing fluxes with the first class of approximations, upwind-biased approximations that model the waves crossing the face the authors introduce the notation FE = nf+L .
• In order to achieve higher than first-order spatial accuracy, an upwind-biased numerical flux.

### F'

• Depends on states other than u;,j,k and U;+l,j,k. Similar calculations using the splitting of Eq. (45) are found in Ref. 6 for an implicit algorithm, and in Refs. 27 and 28 for a two-step explicit algorithm.
• The results of the other two approaches can also be writen in a form analogous to (85).
• Eq. (84) for Steger-Warming flux splitting can be written as (85), with the dissipation term replaced by The other class of higher-order upwind-biased approximations to the numerical flux at a cell face involves the consideration of the wave processes at neighboring faces.
• The local geometric scale of the neighboring face can also be involved.

### Navie r-Stokes Equations

• V S The second form leads to a more complex expression, but is more consistent with the finite-volume philosophy.
• The values of Q in the first two terms, corresponding to the longitudinal component of the gradient, are already given.
• It is interesting to compare these two equations (and the corresponding ones for face if, j , k) with Eqs. ( 79) and (80) that relate the inviscid flux integrals.
• Thus the dependence on the grid geometry due to the transport terms is less compact.
• One also notes that the numerical telescoping of the finite-difference transport flux terms is with adjacent cells.

### Potential Flow

• Another important case where gradients must be calculated is potential flow.
• If one further assumes the flow is isentropic, the.

### MOVING GRIDS

• For an unsteady flow, a grid motion can in general influence the solution of conservation laws in three different ways.
• It affects the convective part of the flux due to the presence of the grid velocity in Eq. (2).
• Grid motion relative to a fixed reference frame has been previously studied by Thomas and Lombard (Ref. 3).
• The effect of a non-inertial reference frame has been treated by Holmes and Tong (Ref. 11) for constant rotation and an explicit integration scheme.
• For this reason, the finite-volume formulation will be given in some detail, and the relevant changes for the finite-difference formulation will be indicated.

### Formulation of General Grid Motion.

• Then the corresponding vector relative to an inertial frame has the general form EQUATION EQUATION ) where C is an orthogonal rotation tensor satisfying.
• The latter may result from the motion of a free surface or the changes in some flow gradients.
• In the most general situation it could depend on all three.
• According to the finite-volume philosophy, the geometric effects due to the grid motion are treated separately from the changes in the physical variables.
• Unless stated otherwise, the authors adopt the notation and EQUATION ) for any quantity x.

### Discretization with Velocity Expressed in Non-Inertial Frame.

• The momentum conservation law for these components requires the presence of source terms.
• But these terms can be eliminated by writing the equations for the Cartesian velocity components instead.
• Before writing the inglicit equation for Afi*, the authors define the tensor and the corresponding matrix operator I" (c-')"+'c" = : x :].
• The scheme is fully conservative, and preserves a uniform free stream.
• For the finitevolume method one must define the location of the cell center when the non-inertial frame undergoes rotation.

### TREATMENT OF BOUNDARIES

• The treatment of flow region boundaries depends on whether conservation is applied to cells defined by the primary or secondary grid.
• These can be conveniently divided into two main classes.
• In one class the unknown variables on the wall are integrated together with those at interior points by an appropriate application of the conservation laws and boundary conditions.
• Both approaches will be discussed for finite-difference and finite-volume grids.
• Note that questions of stability and programming efficiency, although both very important, are beyond the scope of this paper.

### Wall Boundary Conditions for Finite-Difference Grid.

• The simplest way to modify the interior formulas is to replace riJjJo by ri,j,i, and multiply the final result by 2.
• For the second class of methods, the pressure is the only unknown wall quantity needed to calculate flux terms when applying the conservation laws at interior points i, j, 2.
• The practical implementation of all of these approaches could require further approximations which decrease the spatial or temporal accuracy of the algorithm at the boundary, and may involve a restriction to orthogonal grids.
• A factored, implicit, central-differenced implementation of Eq. (139) can only be first order accurate in time due to the presence of Ui,j,2.
• The alternative approach uses the time-differenced forms of the normal momentum equation and Eq. (148).

### t roducing

• The treatment of wall boundary conditions for finite-volume grids basically follows that described for finite-difference grids, with Eqs. (139) and (142) replaced by Eqs. (154) and (157), and the finite-difference boundary point i , j , 1 replaced by the finite-volume boundary point i , j , i.
• Note that conservation of all conservative variables for the entire flow region is automatically satisfied when a finite-volume grid is used.
• All these factors make an implicit algorithm more involved near the boundary.
• The second method of treating the boundary is obviously the natural one for a finite-volume grid.
• The normal momentum equation based on Eq. (157) can be utilized to various degrees of approximation for this purpose.

### Zonal Boundaries.

• Another situation where the difference in the type of grid is important is the case where the boundary is a zonal boundary between two regions with completely disparate grids.
• The partitioning of the flux can now be carried out directly on the zonal boundary, leading to a conceptually simpler algorithm.
• Calculations using a finite-volume grid have been carried out by Eriksson and Rai (to be published) and Walters et a1 (Ref. 48).
• The MUSCL approach, which was also used in Ref. 48, is probably the best one in this situation.
• Note that boundary faces for three-dimensional zonal boundaries will no longer be quadrilaterals in general, and expressions for a general polygonal face must be used to calculate surface area vectors.

### Grid Singularities.

• There are two types of grid singularities that can occur on a boundary.
• For a finite-volume grid these singular points are cell vertices, and their singular nature will not affect the evaluation of geometric quantities based on straight line connections.
• The authors illustrate the procedure for a Navier-Stokes central-difference algorithm near an H-type singularity in two-dimensional flow.
• One must therefore use the above concepts to modify the numerical grid generation scheme.

### STRONG AND WEAK CONSERVATION

• The analysis in this paper has been carried out using vector notation, even though all computations are ultimately performed in terms of scalar components.
• Secondly, there are a number of different ways to obtain scalar equations from the vector equations.
• This terminology was coined by this author in his earlier paper devoted to the differential formulation of conservation laws (Ref. 1).
• Various methods of obtaining scalar equations in strong conservation-law form are discussed in Ref. 1.
• Thus the integral conservation law can always be satisfied for these cases.

### Quasi-One-Dimensional Flow.

• The differential formulation of quasi-one-dimensional flow results from applying the coordinate transformation z( [) to a one-dimensional channel whose crosssectional area vector is EQUATION ) where i is the unit vector in the x-direction.
• In terms of the cell volume V = Szt and the velocity component u = i-u, the continuity and energy equations for inviscid flow become.
• The momentum equation is usually written as The source term pSt results from the pressure acting on the channel walls.

### Ref. 51) write the momentum equation in the quasi-conservative form

• In order to circumvent the weak conservation-law form, some investigators (see in which differentiated terms are multiplied by geometric quantities.
• According to the quasi-one-dimensional approximation, this relation is Pwi = Pi. (176) Thus the undifferentiated term only appears to be an interior source term when relation (176) is used as an ezaet identity to eliminate ptu, in Eq. (175).
• The strong conservation-law form (20) with P = 0 and the derivative term absent results for both flows, except for the momentum equation.
• The proper way to discretize the equations is to apply the conservation laws to primary grid cells.
• The axis of symmetry then serves as a boundary of zero surface area for a row of wedged-shaped cells, and consequently does not contribute to any flux calculations for those cells.

### CONCLUDING REMARKS

• This survey of finite-difference and finite-volume approaches has revealed that comparisons must be made on two levels.
• These affect questions of accuracy and programming efficiency, but are not of a fundamental nature.
• Any boundary procedure can be adapted to either type of grid.
• The changes in conservative variables are then rezoned in a conservative manner to yield the changes in the secondary cells.
• There is a superficial resemblance between the finite-volume and finite-element methods, and much semantic confusion in the literature between the two concepts.

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