Energy Stability nalysis of Multi-Step Methods on Unstructured Meshes by Michael Giles CFDL-TR-87- March 987 This research was supported by ir Force Oce of Scientic Research contract F4960-78-C-0084, supervised by Dr.. Wilson.
Contents Introduction Energy nalysis of Convection Equation 4 3 Semi-discrete nalysis 6 4 Fully Discrete nalysis 7 4. Two-stage time discretization........................ 7 4. ameson's four-step method......................... 9 5 Cell-based Dierencing 0 5. Basic denitions............................... 0 5. Property................................... 5.3 Property 3................................... 5.4 Extension to 3-D............................... 3 6 Node-based Dierencing with Triangular/Tetrahedral Cells 4 6. Basic denitions............................... 4 6. Equivalence to cell-centered dierencing.................. 4 6.3 Extension to 3-D............................... 5 7 nalysis of Systems of Equations 7 7. nalytic equations.............................. 7 7. Semi-discrete equations........................... 8 7.3 Discrete equations.............................. 8 7.4 Euler equations................................ 0
Introduction In an eort to avoid the geometric limitations of structured, logically rectangular meshes, much work is now being done using unstructured meshes for ow calculations around complex geometries. In particular, triangular meshes are becoming popular for twodimensional problems with multiple components [] or grid adaptation [], and tetrahedral cells are being used for three-dimensional aircraft calculations [3]. It is clear that these unstructured meshes will be of increasing importance in computational methods. During this period of rapid development the development of methods for numerical analysis has lagged behind. Questions about the maximum stable time step for explicit methods have been resolved by ad hoc procedures which are based upon the results for structured meshes and a clear understanding of the physical domain of dependence restrictions which are the underlying origin of these limits. The lack of rigorous analysis tools however leads to concerns that current methods may be overly conservative in order to ensure stability in all possible cases. s more three-dimensional calculations are being performed for engineering design and analysis, larger time steps and faster convergence rates can lead to signicant savings in time and money. Thus it is important to develop new methods of analyzing numerical methods on unstructured meshes. The standard approach for analyzing methods on structured grids is to use Fourier analysis, by considering a general solution to be a sum of Fourier modes, and then separately analyze each one [4]. The key point is that on a regular innite grid the eigenmodes are always Fourier modes. On irregular unstructured grids this is clearly not the case and so a dierent approach is needed. The answer is to turn to another classic analysis method, the energy method [4]. This technique denes an energy associated with a solution and then proves stability by showing that the energy is non-increasing. For structured meshes this approach can be much more cumbersome than Fourier analysis, particularly for systems of equations where it can be dicult to pick the correct energy denition, but it is very suitable for unstructured meshes. In this paper we demonstrate the use of the energy method to analyze multi-stage methods for solving the model convective equation on unstructured meshes, using cellbased and node-based spatial dierencing, both of which were developed by ameson [5, 3]. Cell-based dierencing (in which the variable is assumed to be constant within each cell) was the rst developed, but is only rst-order accurate in an integral sense on irregular meshes. In node-based methods the variable is dened at each node, and in general the solution is second order accurate. In performing the energy stability analysis
we will assume that all functions (and their derivatives where necessary) are zero on the boundaries, to avoid the complications introduced by boundary contributions. The inclusion of the boundary terms in the analysis would bring us into the subject of the stability analysis of numerical boundary conditions which is an additional subject in its own right and is beyond the scope of this present paper. Here we are simply concerned with nding the requirements for the interior numerical scheme to be stable. few comments are appropriate on the organization of this paper, since it may seem strange that unstructured grids and spatial dierencing are not discussed until the fth section. Section presents the use of the energy method to prove stability for the analytic convective equation and introduces the ideas and formalism which will be used for the discrete methods. Section 3 analyzes semi-discrete methods (spatially discrete but continuous in time) in a very general form and section 4 extends this to fully discrete equations. These two sections assume that the spatial discretization has certain properties, and then the next two sections prove that the cell-based dierencing and the triangular/tetrahedral node-based dierencing do in fact have these properties. The reason for this approach is to emphasize the importance of these properties, in that any scheme satisfying these conditions will be stable. Finally, the last section extends the analysis to systems of equations and analyzes the Euler equations in particular. 3
Energy nalysis of Convection Equation The model convection equation is @u @t + @u 0: () The `energy' of u(~x; t) in a three-dimensional volume V is dened by E(t) The rate of change of energy is given by de dt Z V Z u (~x; t) dv: () V Z Z V Z u @u @t dv V @V u @u dv @ (u ) dv u n x ds: (3) The last integral is over the surface of the volume with n x being the x-component of the outward pointing normal. If u 0 on @V then the boundary contribution disappears and we are left with de dt 0; (4) proving that the convection equation is energy-preserving and thus stable. We now repeat this analysis using the formalism which will simplify our later analyses. We begin by dening a scalar product, which is a generalized dot product of two functions, and a norm, which is a generalized magnitude of a function [6]. (u; v) kuk Z V q uv dv (5) (u; u) (6) scalar product and norm have a number of basic properties. 4
linearity (u; v + w) (u; v) + (u; w) symmetry (u; v) (v; u) zero norm u 0 ) kuk 0 (7) positive norm u 6 0 ) kuk > 0 triangle inequality ku + vk kuk+kvk In addition this particular scalar product also has the following property. Property u; @v @u + ; v 0 This comes from the divergence theorem with an assumption that u and/or v is zero on @V so that we can ignore the boundary contributions as discussed in the introduction. n immediate deduction from Property is that u; @u dt u; @u u; @u @t + @u ; u 0: (8) With the energy dened by E kuk, the energy analysis is now almost trivial. de + u; @u @t u; @u @u @t ; u 0 (9) 5
3 Semi-discrete nalysis In this section we consider a function u i (t) which is spatially discrete but continuous in time. We assume that we have some discrete spatial dierencing operator @ x so that @ x u is a discrete approximation to @u. The semi-discrete approximation to the convection equation is then given by @u i dt +(@ xu) i 0 (0) To perform the energy stability analysis we assume that we also have a scalar product (u; v) which has the property Property (u; @ x v) + (@ x u; v) 0 s with the analytic version of Property, an immediate corollary is (u; @ x u) 0 () The stability proof is now exactly the same as for the analytic equation. de dt d dt kuk u; du dt u; du dt du + dt ; u (u; @ x u) 0 () The proof is so simple because all of the hard work is hidden in determining the properties of the dierencing scheme, which is particularly involved on irregular grids. 6
4 Fully Discrete nalysis 4. Two-stage time discretization The two-step predictor/corrector method is given by u i u n i t@ x u n i (3) u n+ i u n i t@ x u i u n i t@ x u n i +t@ x t@ x u n i (4) The reason that the ordering of t and @ x is kept as it is in the last equation, is because a very useful technique for accelerating convergence to steady state solutions is to use `local time steps' in which t varies over the domain. In order to analyze this possibility t and @ x cannot be interchanged because the ordering produces dierent results. Because of the variable time steps we dene a new generalized energy E n kt m u n k (5) Note: m is an exponent whereas n is a superscript denoting an iteration time level. We also need to assume an additional property. Property (e i ; e j ) 0 if i 6 j, where e i is a function which has value at node i (or in cell i) and value 0 elsewhere. There are two important corollaries that arise from this assumption. If s is a scalar function then using the linearity of the scalar product it follows that (su; v) i;j i i i;j s i u i v j (e i ; e j ) s i u i v i (e i ; e i ) u i s i v i (e i ; e i ) u i s j v j (e i ; e j ) (u; sv): (6) The reason that this refers to s i being scalar is that we are leaving open for the future the possibility that u i and v i are vectors, in order to analyze systems of equations. 7
The second corollary is similar to the rst. If s and t and two scalar functions, and js i j < jt i j for each i, then ksuk i;j i i i;j s i u i s j u j (e i ; e j ) s i u i (e i ; e i ) t i u i (e i ; e i ) t i u i t j u j (e i ; e j ) ktuk: (7) We now proceed with the stability analysis as before. t m u n+ kt m u n k (t m (u n t@ x u n +t@ x t@ x u n ) ; t m (u n t@ x u n +t@ x t@ x u n )) (t m u n ; t m u n ) t m+ @ x u n t + m+ @ x t@ x u n + t m u n ; t +m @ x t@ x u n t m u n ; t +m @ x u n t +m @ x u n ; t +m @ x t@ x u n t m+ @ x u n t + m+ @ x t@ x u n + t +m u n ; @ x u n t +m u n ; @ x t@ x u n t +m (t@ x u n ) ; @ x (t@ x u n ) (8) The rst corollary was used in the above equation to `switch' the time step terms from one side of the scalar product to the other. In order to eliminate the last two terms using Corollary, we now choose m. The third term can be rearranged using Property to obtain u p n+ t u p n t p t u n + p t t@ x u n Thus the method is stable provided (9) p t @ x p t v kvk (0) for all v. We now introduce the last assumed property for the spatial discretization. Property 3 for all v. There exists a function (t max ) i such that p t max @ x p tmax v kvk 8
Given this property, then provided t i (t max ) i it follows that p t @ x p t v p p tmax @ x t v p p tmax @ x tmax s t t max v s t t max v! kvk () and so the two-step method is stable. The second corollary of Property was used to obtain two of the inequalities in the above equation. To obtain steady state solutions as quickly as possible one would use t i (t max ) i but for time accurate calculations requiring a uniform time step one would have to use tmin i (t max ) i. 4. ameson's four-step method ameson's four-step method is u () u n 4 t@ xu n u () u n 3 t@ xu () u (3) u n t@ xu () u n+ u n t@ x u (3) () ) u n+ u n t@ x u n + t@ xt@ x u n 6 t@ xt@ x t@ x u n + 4 t@ xt@ x t@ x t@ x u n (3) Substitution of this into the energy norm, and using the usual methods to eliminate and reduce terms, leads to u p n+ t u p n t 7 + 576 576 p t @ x t@ x t@ x u n p t @ x t@ x t@ x t@ x u n 8 kvk p p t @ x t v (4) where v p t @ x t@ x t@ x u n. Thus it is stable provided t p t max. The same procedure can be used to analyze other multi-stage schemes. 9
5 Cell-based Dierencing 5. Basic denitions Consider a two-dimensional, innite, irregular grid which is composed of polygonal cells C i. Variables are dened to be constant in each cell, so u i is the value of a function u in C i. On the cell face separating cells C i and C j, u is dened to be the average of the values on either side, (u i +u j ) and ~n ij is the unit vector normal to the face pointing outwards from C i, so ~n ji ~n ij. Figure illustrates all of these denitions. If u(x; y) is a continuous dierentiable function, then the mean value of @u C i is given by @u! i i Z C i @u d in a cell i Z un x dl (5) where i is the area of the cell, and the latter integral is around the boundary of C i with n x being the x-component of the outward normal. Hence we dene the discrete dierential operator @ x to be (@ x u) i i un x l (6) The integral has been replaced by a summation over the faces forming with l being the length of the face. This denition is equivalent to that used by ameson et al [5]. The scalar product for this spatial discretization is dened by (u; v) i i u i v i (7) It is obvious that this denition satises the basic requirements for a scalar product and Property. The hard part is to demonstrate that it satises Properties and 3. 0
5. Property By linearity, (u; @ x v) + (@ x u; v) i;j u i v j [(e i ; @ x e j ) + (@ x e i ; e j )] (8) where e i is again the function which is in cell i and 0 elsewhere. Thus it is necessary and sucient to prove that (e i ; @ x e j ) + (@ x e i ; e j ) 0 (9) for all i; j. There are three cases to consider, depending whether i is equal to j, and if not whether i belongs to the set N j of nodes which are neighbors to j (meaning that C i and C j share a common face). a) i j (@ x e i ; e i ) (e i ; @ x e i ) n x l Z n x dl @C Z i @ () d C i 0 (30) The fact that n x l 0 will be used several times later on in other proofs. b) i 6 j; i6 N j (@ x e i ; e j ) (e i ; @ x e j ) 0 (3) since @ x e j is only non-zero on the cells neighboring C j. c) i 6 j, in j (e i ; @ x e j ) + (@ x e i ; e j ) n x ij + n x ji 0 (3) since the outward normals to each cell are in opposite directions. This completes the proof that cell-based dierencing satises Property.
5.3 Property 3 We need to prove that there exists a function t such that for all v. p t @ x p t v kvk (33) Let (t) i i jn x jl (34) Then p t @ x p t v i i t i i @ x p t v 0 t i i On the face shared by cells C i and C j, p t v p t vnx l pt v i n x l + @ p t v nx l @Ci pt v i + pt v pt v jn i j (35). Hence j (n xl) ij (36) The rst sum is zero because n x l 0. Substituting this equation into Equation (35) we nd that the contribution due to cell C i is 0 t i @ p t vnx l t i pt pt 4 v v (n x l) ij (n x l) ij i j j @Ci 4 4 4 i t i i t i i j t i i j p j t v p j t v jn x lj ij jn x lj ij pt + v jn x lj ij jn x lj ij j j j j pt v j j 4 j 3 jn x lj ij 5 4 j 3 t v jn x lj ij 5 j t v jn xlj ij (37) j jn i The key result which is used in establishing the inequality in the above equation, is that for any pair of real numbers f and g, (f g) >0 ) jfjjgj< (f +g ). Summing
over all of the cells we nally get p t@ x p t v i j t v j jn i t v j jn xlj ij in j jn x lj ij j j v j kvk (38) This completes the proof that the cell-based dierencing satises Property 3, with (t max ) i i jn x jl (39) Not only does this give a sucient condition for stability (when combined with the theory of the last section); it also gives the necessary condition for regular quadrilateral. Using Fourier analysis it can be shown that for a uniform grid of parallelograms the stability limit is where 8 < : t t max (40) two-step method p four-step method and t max y where is the area of the cells and y is dened in Figure. This equation for t max is exactly the same as is given by the present theory. 5.4 Extension to 3-D The extension to three-dimensional grids is actually very straightforward. ll that changes is that cell areas i become cell volumes V i, and face lengths l become face areas. With these minor changes all of the theory and the proofs carry over directly (although it becomes much harder to visualize some of the steps involved). The discrete dierential operator and maximum time step are (@ x u) i V i (t max ) i V i u n x (4) jn x j (4) 3
6 Node-based Dierencing with Triangular/Tetrahedral Cells 6. Basic denitions In node-based schemes the variables are dened at the nodes of the computational cells, and because of a critical step in the proofs to be presented we only consider triangular cells. ssociated with each node i is the `supercell' Ci 0 formed by the union of the triangles with node i, as shown in Figure 3. The discrete dierential operator @ x is dened by (@ x u) i 0 i @C 0 i u 0 n 0 xl 0 (43) with u 0 on a face dened as the average of the values at the two nodes at either end. The scalar product is dened by associating of the area of each triangular cell with 3 each of its nodes. (u; v) i 3 iu i v i (44) gain it is obvious that this denition satises the basic requirements for a scalar product and Property. 6. Equivalence to cell-centered dierencing To prove that the node-based dierencing satises properties and 3, we will show that the node-based dierencing is equivalent to a cell-based dierencing on an alternative cell with a modied area, and hence the proofs of the last section are equally applicable to node-based dierencing. Figure 3 shows the alternative cell C i for the cell-based dierencing, which is formed by joining the centroids of the triangles surrounding node i. The important geometric relation is that ~x a 3 (~x i + ~x j + ~x k ); ~x b 3 (~x i + ~x j + ~x l ) (45) ) (l~n) ij 3 Thus the contribution of node j to (@ x u) i is 0 i h l~n 0 kj + l~n 0 jli u j h l~n 0 kj + l~n 0 jli 4 (46) 0 i 3 (ln x) ij u j (47)
since ) (@ x u) i 0 i 3 jn i 0 i 3 jn i 0 i 3 u i (n x l) ij u i jn i u j (n x l) ij (u i+u j ) (n x l) ij u n x l (48) n x l 0 (49) Thus the denition of @ x u for node-based dierencing on Ci 0 is identical to the denition of @ x u for cell-based dierencing on C i, except that it uses 3 0 i instead of i. For regular grids these two are equal but in general for irregular grids they will not be equal. Examining all of the proofs in the last section on cell-based dierencing, the fact that i is the area of the computational cell is not required in any of the proofs, and so any value for i can be used provided that value is consistently used in the denition of @ x, the scalar product and t max. lso interesting is that when using local time-steps the important expressions are p t max u and t max @ x u, and in both cases the area terms cancel out leaving only u i jn x jl and u n x l jn x jl respectively. Thus the cell area plays a i very minor role and the important terms are due to the faces. Returning to the node-based dierencing, all of the proofs for properties and 3 apply directly with t max 3 0 i jn x jl (50) It is interesting that the maximum time step depends on the faces for the equivalent cell-based algorithm. It suggests that in some sense the cell-based dierencing is more natural or more basic. 6.3 Extension to 3-D The extension to 3-D is quite natural, with the cells for the equivalent cell-based algorithm being constructed by joining the centroids of the tetrahedra. by (@ x u) i V 0 i @C 0 i u 0 n 0 x 0 5 node-based @ x is dened
4 V i 0 u n x cell-based (5) and t max 4 V i 0 jn x j (5) 6
7 nalysis of Systems of Equations 7. nalytic equations To extend the preceding analyses to rst order systems of equations we begin by considering the following vector equation. @u @t + @u + B @u @y 0 (53) u is now a vector of dimension m and and B are constant m m matrices. The scalar product is now dened as (u; v) Z V u T v dv: (54) Property is still satised since + u; @v @u ; v Z Z V @V @ u T v dv u T v n x d 0 (55) provided we ignore boundary contributions. lso it is clear from the denition of the scalar product that (u; v) T u; v : (56) Using these results the energy stability analysis proceeds as follows. de u; @u @u + dt @t @t ; u u; @u @u +B @u @u +B @y @y ; u u; @u u; B @u @u @u @y ; T u @y ; BT u u; T @u u; B T @u B @y (57) For de dt to be zero for all possible u requires that and B both be symmetric. For the present purposes we are only interested in hyperbolic, energy-preserving systems and so we will assume that this is the case. 7
7. Semi-discrete equations The semi-discrete analysis assumes that for any symmetric matrix (u; v) (u; v); (58) in addition to the usual basid properties and Property. It is clear that these are all satised by the scalar product denitions for both cell-centered and node-centered dierencing in which (u; v) i u T i v i i (59) The non-bold i is the cell area as dened earlier. corollary from this assumption is that (u; @ x u) [(u; @ xu)+(u; @ x u)] 0; (60) (u; B@ y u) [(u; @ ybu)+(bu; @ y u)] 0: (6) Hence the stability analysis of the semi-discrete equation, is simply de dt du dt + @ xu + B@ y u 0; (6) u; du dt (u; @ x u) (u; B@ y u) 0: (63) 7.3 Discrete equations The analysis of the fully discrete equations also proceeds almost exactly as before. Omitting the tedious algebra, the nal result for the two-step method is p u n+ t u p n t and so it is stable provided + p t (@ x +B@ y ) n u p t (@ x +B@ y ) t (@ x +B@ y ) u n (64) p t (@ x +B@ y ) p t v kvk (65) 8
for all v. For the cell-centered dierencing, this is true if t i < (t max ) i where (t max ) i i The matrix norm jcj is dened by jcj max v jn x +Bn y jl: (66) jcvj ; (67) jvj so that jcvj jcjjvj. If C is symmetric then the norm is equal to the absolute magnitude of the largest eigenvalue of C. The proof that the given denition of t max is sucient is again very similar to the scalar proof. p t (@ x + B@ y ) p t v i i i i i i i i j j t i i (n x +Bn y ) p t v l t i q t j (n x +Bn y ) 4 ij v j l ij i jn 8 i t i < T q t 4 i : j (n x +Bn y ) ij v j 8 t i < 4 i : q j N i j N i j N i 8 t i < 4 i : 8 t i < 4 i : 8 t i < 4 i : t j (n x +Bn y ) ij v j l ij l ij 9 ; 9 q t j (n x +Bn y ) ij v q j t j (n x +Bn y ) ij v j lij l ij j ; N i 9 q t j jv j j t j jv j j jn x +Bn y j ij jn x +Bn y j ij l ij l ij ; q j N i j N i j N i j N i 9 t j jv j j +t j jv j j jn x +Bn y j ij jn x +Bn y j ij l ij l ij ; 9 8 < 9 ; : ; j N i jn x +Bn y j ij l ij jn i t j jv j j jn x +Bn y j ij l ij t j jv j j in j jn x +Bn y j ij l ij i jv j j j N i t j jv j j jn x +Bn y j ij l ij kvk (68) 9
7.4 Euler equations barbanel and Gottlieb [7] have shown that with an appropriate choice of variables the linearized Euler equations can be written in the above rst-order form with symmetric matrices and B equal to 0 q u c 0 0 B @ q q c u 0 c 0 0 u 0 q 0 c 0 u C ; B 0 B @ q v 0 q c 0 0 v 0 0 q c 0 v q c 0 0 c v With these denitions it is a straightforward exercise to show that the eigenvalues of n x +Bn y are ~u:~n, ~u:~n, ~u:~n c and ~u:~n+c, where ~u:~n is the normal ow velocity and c is the speed of sound. Thus it follows that for the cell-based dierencing the time step limit is given by C (69) (t max ) i i (j~u:~nj+c) l (70) It can be shown that for low Mach number ow over a regular square mesh this `maximum time step' is a factor p smaller than the value obtained by Fourier analysis, but they are equal in the limit of high Mach number or high cell aspect ratio or high cell skewing. Thus for practical purposes it is a sucient and almost necessary condition for numerical stability. s before the theory extends easily to three dimensions, and for the Euler equations the resultant time step limit for cell-based dierencing is (t max ) i V i (j~u:~nj+c) (7) 0
References [] D. Mavriplis and. ameson. Multigrid solution of the Euler equations on unstructured and adaptive meshes. In S. McCormick, editor, Proceedings of the Third Copper Mountain Conference on Multigrid Methods: Lecture Notes in Pure and pplied Mathematics. Marcel Dekker Inc, 987. [] R. Lohner, K. Morgan,. Peraire, and O.C. Zienkiewicz. Finite element methods for high speed ows. I Paper 85-53, 985. [3]. ameson, T.. Baker, and N.P. Weatherill. Calculation of inviscid transonic ow over a complete aircraft. I Paper 86{003, 986. [4] R.D. Richtmyer and K.W. Morton. Dierence Methods for Initial-Value Problems. Wiley-Interscience, nd edition, 967. Reprint edn (994) Krieger Publishing Company, Malabar. [5]. ameson, W. Schmidt, and E. Turkel. Numerical solutions of the Euler equations by nite volume methods with Runge{Kutta time stepping schemes. I Paper 8-59, 98. [6]. Mathews and R.L. Walker. Mathematical Methods of Physics. W.. Benjamin, Inc., 970. [7] S. barbanel and D. Gottlieb. Optimal time splitting for two and three dimensional Navier-Stokes equations with mixed derivatives. ICSE Report No. 80-6, 980.
P PPPPPPPPPP C j 6~n ij l ij - C CC? ~n ji C CC C i C````````` Figure : Geometric denitions for cell-based dierencing rea 6? y Figure : Denition of y for regular skewed mesh
~n 0 kj j K ~n ij k a PP * @ P PPP @ b @ @ @ l i P PPPPPPPPPPPPPP P ~n 0 jl @C 0 i Figure 3: Geometric denitions for node-based dierencing 3