A Finite-Volume Discretization of the Shallow-Water Equations in Spherical Geometry

Transcription

1 A Finite-Volume Discretization of the Shallow-Water Equations in Spherical Geometry Lars Pesch Meteorologisches Institut der Universität Bonn Auf dem Hügel 20, Bonn February 12, 2003 Abstract A numerical method for the solution of the shallow water equations in spherical geometry has been developed. Special emphasis is laid on respecting conservation properties in the discrete solution. Therefore the spatial discretization uses a finite-volume method, which is applied on a spherical geodesic grid generated by recursive refinement of an icosahedron. The potential vorticity streamfunction formulation of the equations is used and several methods for the iterative solution of the resulting Poisson equations are examined. Based on a set of standardized test cases the resulting model s performance is investigated. The conservation properties are among the strengths of the method, which are discussed together with its weaknesses and possible improvements. 1 Introduction The idea for this project was born from a close examination of results of the German Weather Service s (DWD) operational global forecasting model (GME). In model integrations of GME integral invariants of the atmospheric system like mass or energy are not preserved. The reason is the use of nonconservative finite-difference operators for discretizing the equations of fluid motion. On the other hand GME offers some in- Currently at: Department for Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, UK; L.Pesch@damtp.cam.ac.uk teresting features for example its icosahedral grid which have made it a candidate for being the base of a climate model yet to be developed, a successor of ECHAM 5. In the context of climate modeling however, violation of conservation laws is even less acceptable than in short- or medium-term weather forecasting. With these guidelines the idea was to test the applicability of methods that are able to maintain conservation properties in the discrete solution. For equations in conservation form finite-volume techniques, whose basic ideas are summarized in Section 3, are the method of choice. The common approach is to test new methods on a simplified set of equations before developing them further until they can possibly be used in a weather or climate model. A suitable set for this purpose are the shallow-water equations (SWE), which will be presented in Section 2. The desire to conserve integral invariants also lead to the use of a formulation of the SWE that does not directly predict the primitive variables; the equations can be rewritten using potential vorticity and divergence as prognostic variables, which is an attractive alternative as will be shown in Section 2. Some of the problems that emerge during the numerical treatment of the prognostic equations are founded in the choice of grid (i.e. the distribution of grid points in which the variables are defined) and/or coordinate system. These are overcome in GME by the help of a gridding procedure that starts from an icosahedron a threedimensional object composed from 20 equilat-

2 2 3 FINITE-VOLUME METHODS eral triangles and increases the number of grid points by recursive refinement of the triangles. This results in a relatively even distribution of the grid points over the globe in contrast to the geographic grid with its typical concentration of grid points near the pole. Section 4 contains information about this grid. The time stepping scheme that has been used is explained in Section 5. The discretization of the prognostic and diagnostic equations is described in Sections 6 and 7, respectively. Numerical results are presented and discussed in Section 8. A short summary and an outlook on possible developments conclude the article. 2 The Shallow-Water Equations The SWE result when simplifying the Navier- Stokes equations using the following assumptions for large scale atmospheric motion: inviscid and hydrostatic flow of an incompressible homogeneous fluid whose vertical extent is small compared to the horizontal size of the domain; in fact the equations are integrated over the height of the fluid layer and therefore the resulting flow description is two-dimensional. While exhibiting an easier mathematical structure than the complete system of equations, the SWE still capture most of the horizontal dynamics of the atmosphere. Thus they provide a suitable first test which a new scheme has to master before being applied to the baroclinic equations. The conservative or flux form of the SWE is given by the following system: h + (hv) = 0. (1) t hv + (vhv) = gh (h s + h) fe r hv t (2) where h is the thickness of the fluid layer, h s the height of the surface topography, v the horizontal velocity, e r the outward unit normal vector on the sphere, g the acceleration of gravity and f the Coriolis parameter. The formulation of the SWE using h together with potential vorticity (PV) q and divergence δ as prognostic variables offers a second conservative variable: PV. Consequently, the prognostic equations for the first two variables, which can be written in flux form, display a mathematical structure which is ideal to use with finite-volume methods. h + (hv) = 0 (3) t (hq) + (hqv) = 0 (4) t [ δ t + hqe r v ( + g(h + h s ) + v v 2 ) ] = 0. (5) Another aspect that counts for this set is the fact that the model variables are all scalars and not vector components. Using the equations (3), (4) and (5) to forecast the behavior of the fluid dynamical system a problem arises due to the under-determined system of equations: the fluxes that occur in the equations contain the velocities v, and these are no longer described by prognostic equations. Instead, they have to be retrieved from the derived quantities that are now modeled prognostically. A common way to determine the velocity is to solve two Poisson equations for the streamfunction ψ and velocity potential χ: 2 ψ = ζ = hq f (6) 2 χ = δ. (7) This method will also be used here. The velocity field is subsequently determined by differentiation: v rot = e r ψ (8) v div = χ (9) v = v rot + v div. (10) 3 Finite-Volume Methods Finite-volume (FV) methods start by partitioning the domain of the flow into a set of cells C. These cells are the smallest unit on which information about the functions or variables is available. The consistency requirement for the cells is that they do not overlap but cover the whole domain. The evaluation of the balance

3 3 equation is then carried out on a set of controlvolumes V. Volumes may overlap partly, in fact in some cases one is forced to use overlapping volumes, for instance when there are more nodes than cells 1. Conservation is easily maintained as FV methods are used together with the integral form of the governing equations. For a quantity ψ the flux form is integrated over a control-volume d dt V ψ dv + V n F ψ ds = V Q ψ dv, (11) with the reference volume V, the flux density F ψ and a source density Q ψ. The balance of the modeled quantities on these volumes is evaluated by integrating the fluxes entering and leaving each volume to find the net flux. By performing only a single evaluation of each flux over a volume face and using it for the update on both sides of the boundary, the integral of the quantity over the two neighboring volumes will be unchanged after the update, as long as the integral of the source term vanishes, too. In our case equations (3) and (4) are free of source terms unlike the equations for the momentum components. The divergence equation includes a source term within the divergence term; its treatment will be shown later. FV methods are used in many CFD simulations and have been developed to reach e.g. higher order accuracy and other properties. Details about these aspects can be found in many CFD books. 4 The Icosahedral Grid Mostly the geographic grid is considered to be the natural grid on the sphere probably for reasons of its common use (also in geographic maps) and its manageability when discretizing differential equations on the sphere. The problems arising from this approach have been known for a long time. Alternative methods have been discovered focusing on the uneven distribution of the grid points over the sphere and the use of a single coordinate system. Some 1 Note for example that the triangular grid that originates from refinement of the icosahedron has about twice as many cells (triangles) as nodes. techniques start from the platonic solids and use these in a certain sense very regular objects to yield a refined grid with relatively smooth grid properties. The method used here (based on the refinement of the triangular grid of an icosahedron) was developed by Baumgardner (1983) and Baumgardner and Frederickson (1985). An icosahedron has twenty equilateral triangular faces, five of which join in each of the twelve corners that are connected by thirty edges. When embedded into a sphere the edges can be projected onto the sphere so that it will be partitioned into a spherical icosahedron. The spherical icosahedron serves as a macrotriangulation of the sphere which allows the application of common refinement techniques to obtain a denser grid. In order to keep the grid distortion at a minimum, the refinement method chosen here is the following: in every step each midpoint of a triangle edge is connected with the midpoints of the two neighboring sides of the triangles on both sides of it. Like this the grid remains triangular and every grid point has six neighbors with the exception of the corners of the original icosahedron which have only five. The first three grids are shown in Figure 1; a few of their characteristics can be found in Table 1. The number ni states into how many parts an edge of the original spherical icosahedron has been divided and it is the measure for the grid resolution that will be used in the following. In order to apply a finitevolume method we will not primarily use this triangular grid. The reason is that the method models the change of integrated quantities on test volumes. On the triangular grid however we have about twice as many triangles (equations) than nodes (degrees of freedom), so that the discrete system would be overdetermined. A different aspect is that on triangular grids neighboring cells are not necessarily connected by an edge. Thus information cannot flow from one cell to all its neighbors within one time step. The method to overcome this difficulty is to define a so-called dual or Voronoi grid. Here, the dual grid is constructed as shown in Figure 2: In each spherical triangular cell we find the center of mass Q i (in fact we project the center of mass of the flat triangle with the same corners onto the unit sphere) and connect these points with

4 4 4 THE ICOSAHEDRAL GRID ni N 10,242 23,042 40,962 92, , , ,362 T 20,480 46,080 81, , , ,280 1,310,720 max (km) min (km) T42 T63 T106 Table 1: Number of grid points (N = 10 ni 2 +2) and triangles (T = 20 ni 2 ), maximal and minimal distance ( ) between two grid points at different resolutions of the refined icosahedral grid. The last line shows approximately corresponding spectral truncations. P 5 P 4 Q 4 Q 5 Q 3 P 6 P P 3 0 Q 6 Q 2 Q 1 P 1 P 2 Figure 2: Definition of the dual grid. Figure 1: The first refinements of the icosahedral grid: ni = 1 (original icosahedron), 2 and 4. geodesic arcs. The resulting hexagon/pentagon around each node of the triangular grid is its associated dual cell. Its area can be calculated by dividing it into the appropriate number of spherical triangles including two centers of mass Q i, Q i+1 and the central node P 0. The icosahedral grid has provided a smooth distribution of the grid points over the sphere and thus solves one of the two problems which have been mentioned before. Another problem is encountered when using a single geographic coordinate system which is singular at the poles. In order to avoid the pole singularities and to be able to represent the tangential velocity in each grid point P by only two components, local spherical coordinate systems are defined in every grid point. This idea goes back to Baumgardner and Frederickson (1985), who derived the differential operators in these coordinate systems and thus avoided the pole problem at the cost of coordinate transformations. The basis vectors of the local systems are denoted by e λ pointing to the local east, and e ϕ aligned with the north direction. They define the coordinate lines of η and χ, the coordinates of the local spherical system. In its own local system, the grid point P is situated at the position η = χ = 0. With this ansatz the local system is almost Cartesian in the cell: the poles of the local coordinate system are far away compared to its neighboring grid points. As the components of all vector quantities located in P are stored as coefficients of these basis vectors, they as well as the coefficients of vectors located in neighboring grid points have to be transformed into a common system whenever using them in a vector computation. Details about the local coordinate system, the determi-

5 5 nation of the local basis vectors and transformation weights can be found in Majewski (1996). Besides these advantages the icosahedral grid facilitates the implementation of numerical methods as it can be decomposed into a manageable data structure by grouping the icosahedron s triangles into ten rhombic structures (these are called diamonds) which possess a twodimensional, rectangular data structure. Extra care has to be taken where the diamonds abut (especially at their corners) because there the addressing of neighbouring grid points differs from the normal (and constant) relationships on the rest of the diamond. Icosahedral discretizations are not free of problems. E.g. it can be detected in long term runs that the fivefold symmetry of the icosahedron carries on into the error patterns of solutions for the discretized differential equations. Research is under way to reduce these effects, for example by breaking the symmetry pattern of the underlying regular object by a modification allowing the grid points to float around their original position so as to minimize a suitable measure for the truncation error of the discretized operators (Tomita et al. 2001). 5 Time Integration: The Adams-Bashforth Schemes Apart from the spatial dimensions the equations have to be discretized with respect to time, too. This was accomplished by one of the Adams- Bashforth schemes, which belong to the family of linear multistep methods (LMMs), i.e. they employ data from more than one previous time level to compute the change during a time step. The nonlinear ordinary differential equation (i.e. in our case the partial differential equation after spatial discretization) of the form du dt = F (u, t) (12) has the K-step LMM discretization U n+1 = U n + t F n (13) U n+1 = U n t F n (14) U n+1 = U n + t [ 2 3 F n F n 1] (15) U n+1 = U n + t [ F n 16 F n F n 2] (16) which are the Euler (13), Leapfrog (14) and Adams-Bashforth second (15) and third order (16) methods, respectively. The higher order of accuracy in the third order scheme (AB3) is achieved by using the data from two previous time levels (except the current level n). The additional computation time is moderate as the tendencies can be stored for each cell and just need to be combined according to (16). The main effect is the increased memory requirement, but memory size is usually not a limiting factor in today s computer systems. These are the reasons why the AB3 scheme was used in this model. One might criticize that third order accuracy is not appropriate in the scheme developed here as the spatial discretization is only of first order. Of course also a lower order method could have been chosen, but the software implementation allowed the third order variant, which has also comparable stability restrictions so preference was given to the higher order scheme. In comparison to the time discretization, unfortunately it is quite difficult to obtain higher order in the FV spatial discretization. Obviously the third-order Adams-Bashforth method (AB3) is not self-starting, the first two steps have to be taken using lower order schemes, e.g. one Euler and one second order Adams- Bashforth (AB2) step. The AB3 scheme is discussed in detail by Durran (1991) and has been applied in the shallow water context by Heikes and Randall (1995a). It may be mentioned that the first implementation of the model used the MacCormack scheme, a method whose development took place together with finite volume methods and which can be found in many textbooks on CFD. As a two-step Lax-Wendroff method however, the MacCormack scheme has approximately doubled runtime requirements in comparison with one-step methods. This is especially disadvan-

6 6 7 DIAGNOSTIC SYSTEM tageous as the potential vorticity-divergence formulation of the SWE requires the costly solution of two Poisson equations in every step. 6 Discrete Prognostic Equations The grid for the discretization has been described until now but a few technical aspects need to be added to complete the description. These will be mentioned in this section before we have a look at results. 6.1 Equations in Flux Form We use the dual icosahedral grid from Section 4 together with a cell-centered FV scheme; in this case, the control volumes V are identical with the cells C. The prognostic equations are integrated over the cells and the volume integrals of flux divergence terms are expressed as line integrals of the flux densities multiplied scalarly with the outward normal vector dr N to the integration path in the local tangential space of the sphere. The numerical value of the volume integrals, i.e. basically C h da is the value at the sole grid point within the volume multiplied with the size of the volume. The line integrals use a numerical flux function F which consists of a physical flux density F and an artificial diffusion flux D: F = F D. The physical flux is computed in the grid points and the values from the two points P across a dual edge Q i Q j are averaged (cf Fig. 2). The artificial diffusion flux is defined directly on the dual edge using differences of the dependent variables across the cell face. The use of artificial viscosity is suggested by Williamson et al. (1992) for the test cases in which the energy cascade towards smaller scales plays an important role and it can dampen the spurious modes in the discrete treatment of the equations. The artificial viscosity terms as they are used here were first applied by Jameson et al. (1981); they take the form of a second and a fourth order term which are present in regions of strong gradients. 6.2 Treatment of the Divergence Equation The divergence equation differs from the other two model equations insofar as it cannot be written in flux form. The difficulty is that in contrast to the flux equations there is still a differential operator in the equation after the integration: t C δ da = C C = C hq(e r v) dr N ( g(h + h s ) + v v ) dr N }{{ 2 } =:τ (17) hq(e r v) dr N C τ n ds. (18) In (18) we have replaced the product n with its simplified meaning: the derivative in the normal direction of the integration path. In the numerical integration this term will be approximated by a finite difference of the two nodal values on both sides of the dual integration arc. 7 Diagnostic System The method to retrieve the velocity components from the model variables potential vorticity and divergence described in Section 2 is not yet applicable with the above techniques as we have no discretization of the gradient operator available through the finite-volume treatment. The Laplace operator has indirectly been discretized for the divergence equation. On the other hand it has to be pointed out that the whole procedure of determining the new velocities is not connected with the conservation properties of the scheme. The conservation of mass and potential vorticity (or any passive tracer) is accomplished by the use of finitevolume techniques in the prognostic equations. The main point is that basically any method for solving elliptic equations can be employed for the first part of the velocity computation. As the vorticity-divergence formulation has been

7 7.1 Discretization 7 used by other authors, it is worth considering their approaches. There is a tendency to exploit the natural hierarchy of refinements of the icosahedral grid by applying multigrid (MG) methods. Heikes and Randall (1995) use a geometric multigrid solver and so does Thuburn (1996). In these articles there are, however, no comparisons of the solver performance with other methods. The main advantage of MG methods is scalability: the rate of convergence and the number of iterations is more or less independent of the problem size for important classes of problems. The above geometric multigrid methods are, however, not used to solve the elliptic problems in the model presented here. Instead, the Poisson equations are discretized at the model resolution and the solution of the linear system of equations can then be carried out by a suitable solver. Like this, flexibility is retained for both these steps and the influence of different components can be investigated. The components of the solution procedure for the Poisson equations are described in the following subsections. 7.1 Discretization The discretization of the Laplace (and gradient) operator has been taken from the original GME, where all operators are discretized with a finitedifference ansatz. The components of the gradient as well as the Laplacian of a scalar field φ in a primary grid point P 0 are formed as linear combinations of the values of φ in P 0 and its direct neighbors P 1,..., P 5 /P 6 surrounding P 0. The values of φ contribute to the desired derivative in P 0 with certain weighting factors G ηi, G χi : φ η = G ηi (φ i φ 0 ) and (19) P0 i φ χ = G χi (φ i φ 0 ) (20) P0 i for the components of the gradient operator in the two (local) coordinate directions η and χ; the Laplacian as the sum of the two second order derivatives is expressed as φ = i L i (φ i φ 0 ). (21) The factors G ηi, G χi and L i depend on the positions of the neighbors in the local coordinate system of P 0 and their derivation can be found in Majewski (1996). In the shallow water integration scheme the problem is not to compute the Laplacian but rather we know the right hand side of the equation φ = f (or L φ = f in matrix-vector notation) and need to determine the vector of unknown values at the grid points on the left hand side φ. As the evaluation of the Laplacian at a grid point P uses the values of φ at the surrounding grid points, now we have to solve a coupled linear system. Unfortunately the system matrix L has one eigenvector with the eigenvalue 0, namely the constant vector whose entries are all the same. Consequently L is singular and no unique solution can be found to the linear system. This is not problematic for the further solution scheme as the resulting field φ will be differentiated subsequently so that the component into the direction of the singular eigenvector is immaterial. However, the singularity does matter for the solution process of the linear system as many solvers rely on regularity or definiteness. L is not symmetric either, which is another problem when using iterative solvers. These issues will be dealt with in the next section. A few alternatives for the components of this procedure shall be mentioned here. If we retained the finite difference ansatz it would be desirable to achieve higher order in the discretization of the Laplace operator. Indeed a second order discretization has been developed by Cassirer et al. (2001) using a larger stencil on the triangular grid. Alternatively a finite-element ansatz could be chosen. Heinze (1998) uses finite elements to solve a Helmholtz equation on triangular elements on the sphere 2. In general however, the problem with the singular matrix at the discrete level remains. 2 The positive constant c in the Helmholtz equation 2 φ+c φ = rhs causes definiteness of the corresponding matrix in contrast to the problem posed here.

8 8 8 NUMERICAL RESULTS 7.2 Choice of a Solver for the Linear System Now that the discrete problem has been formulated a suitable method has to be found to solve the system of equations. As mentioned previously, the singularity of the matrix causes problems. The solver that manages to produce a solution reliably and robustly is the method of generalized minimal residual (GMRES). Like all iterative methods, GMRES should be used together with a preconditioner. Any iterative scheme like Jacobi or Gauss-Seidel relaxation, Successive Overrelaxation (SOR), Incomplete Lower-Upper Factorization (ILU) or multigrid methods can be used as preconditioner. An algebraic multigrid (AMG) method has been tested as a solver component for the Poisson equations as well as ILU and ILUT factorization. 8 Numerical Results To evaluate the scheme developed here it has been applied to the test suite compiled by Williamson et al. (1992); it consists of seven problem settings which increase in complexity and demands on the scheme. Results for some of these test cases will be presented here however first the different solvers for the Poisson equations will be compared. 8.1 Comparison of Solver Configurations The solution of the diagnostic system (6) and (7) is crucial to the performance and quality of the integration scheme. Different methods for the solution procedure have been suggested and some configurations will be compared here. No ideal solution could be found; while the problem can generally be handled successfully, a compromise has to be made regarding runtime requirements on the one hand and quality of the results on the other. The only available measure to evaluate the quality of the solution for the linear system L φ = Figure 3: Averaged residual reduction for different preconditioners, a) ILU, b) ILUT, c) ILU+AMG; the black line is for the system 2 χ = δ, the grey one for 2 ψ = hq f (test case 5, model resolution: ni = 32). f in such cases is the norm of the residual r = f L φ during or after the solver iteration. We will now compare the effectiveness of three GMRES-based solver configurations. For all configurations the dimension of the Krylovsubspace was 20. A smaller dimension might be appropriate, too, but has not been tested. The first two variants are preconditioned by two ILU and ILUT cycles, respectively. The third uses ILU for pre- and postsmoothing and does one AMG cycle in between. In general one can say that the ILU-preconditioned solver needs more iterations than the ILUT or AMG variant but it gives the best results, see Figure 3. The figure shows the residual reduction that is reached after n cycles. The plotted values are averages of 100 time steps at the beginning of the model integration (however the first few steps have not been used). The coarse grid correction with AMG results in a very effective first iteration: the residual is reduced as much as in two to four steps of the one-level methods. Unfortunately after the first iteration the residual reduction practically ceases. It should be clear that the results in Figure 3 are average values and the behavior in a certain time step of the model integration can differ widely. The performance also depends strongly on the flow: Figure 4, for example, shows the results for the same variables taken from a simulation of test case 6. Due to the different dynamical situation the residual of the Poisson-equation for the streamfunction can now be reduced much better, so that the onelevel methods accomplish an improved solution at the cost of more iterations. With the above findings it was also possible to

9 8.2 Test Case 1: Advection of a Cosine Bell 9 initial height field is denoted by h 0 3. For all fur- Figure 4: Averaged residual reduction for different preconditioners, a) ILU, b) ILUT, c) ILU+AMG; the black line is for the system 2 χ = δ, the grey one for 2 ψ = hq f (test case 6, model resolution: ni = 32). compare the runtime requirements necessary for the different components of the model and the various solvers. The by far most expensive component is the solver part for the two systems of equations, which uses 90% to 98% of the total computation time at the resolutions ni = 32 and 64. This effect increases with the resolution as more iterations are needed for the linear solvers. AMG is an exception to the rule, it stagnates after the same amount of iterations in both resolutions. ILU takes about 30% more time than ILUT. In comparison to AMG preconditioning they are slower by a factor of 3 to Test Case 1: Advection of a Cosine Bell This test case does not use the full set of the shallow water equations but only the advection equation with a constant velocity field which describes a solid body rotation, so that the height field is transported without any change of shape. The direction of advection can be changed so that the initial data, a compact cosine bell, takes different paths over the computational grid. The errors allow to evaluate the advective properties of the scheme. A first impression can be gained by plotting the result of the simulation vs. the analytical solution or alternatively the difference between the two. Errors can also be measured in different norms which are defined in the usual way (cf Williamson et al. (1992)). Additionally, the normalized mass is evaluated. If the mean of a quantity h on the sphere is denoted by h the normalized mass is defined as M = h h 0. The h 0 Figure 5: Numerical solution (left column) and error (right column) for test case 1, subcase 1. The grid spacing for the numerical solution is 100 m and the analytical solution is plotted with thin dashed lines. Positive errors are shown by solid lines, negative ones are dashed; the spacing of the error is 50 m. a) + b): ni = 32, c) + d): ni = 64, e) + f): ni = 128. ther results of test case 1 the same small amount of second order artifical viscosity has been added (κ (2) = 0.005). The fourth order diffusion had only a very small effect on the results so that it has been excluded from all tests. The next results that will be presented here concern the improvements of the results when increasing the grid resolution. Subcase 1 (advection along the equator) has been chosen with a fixed time step length and the twelve days were simulated at three resolutions. The results for three resolu- 3 The definition of M is slightly different in Williamson et al. (1992); however the definition M = h h T would h 0 lead to inappropriate results as the volume of the discrete cosine bell varies depending on the position on the diamonds.

10 10 8 NUMERICAL RESULTS tions can be seen in Figure 5. The model is run with the time step length that the CFL-criterion yields: 200s at ni = 32, 100s at ni = 64 and 50s at ni = 128. Most of the error occurs due to the computed solution lagging behind the analytic solution. For ni = 32 the reduction in height of the cosine bell is significant but this improves when higher resolutions are used. The number of oscillations that are generated at the edge of the cosine bell increases with the number of time steps. Of course the conservation property was of central importance when designing the scheme. The results of the normalized mass M for subcase 1 at the resolution ni = 32 are shown in Figure 6. The error is due to the roundoff occurring in the computation; it has no systematic trend. A comparison of the error norms Figure 7: Comparison of the l 1, l 2 and l norms of the height error for test case 1, subcase 1; a) ni = 32, b) ni = 64, c) ni = Test Case 2: Steady State Geostrophic Flow Figure 6: Normalized mass for test case 1, subcase 1 sampled in every time step at the resolution ni = 32, t = 200s. Note that the scale for M is is given in Figure 7, representing the same setting, resolutions and time step lengths as used previously. The error rises almost linearly but the growth rate approximately halves when the resolution is doubled. The results of test case 1 were within the expectations one could have for a relatively simple scheme as it is used here. The reduction of the maximum of the initial data and the introduction of oscillations is a weak point. More elaborate techniques can help to improve this, e.g. shape preserving advection schemes as developed by Thuburn (1996) with results published in (Thuburn 1997). The idea of the second test case is to reproduce a steady state solution of the shallow water equations: zonal flow with the corresponding geostrophically balanced height field. It should be mentioned that the following results have been obtained after initialization with the wind components u and v as given above and not by prescribing as starting values the analytical vorticity and divergence fields. No modifications to satisfy a discrete geostrophic relationship (as mentioned by Williamson et al. (1992)) have been made either. The geostrophic balance is reproduced with some errors. Small deviations from the local geostrophic balance lead to the formation of a Rossby wave with wavenumber five, a manifestation of the fivefold symmetry of the icosahedral grid. This problem is known as the wavenumberfive phenomenon and occurs in many models that use an icosahedral discretization. There

11 8.5 Test Case 6: Rossby-Haurwitz Wave 11 Figure 8: Difference of computed and analytic solution after 5 days for the resolution ni = 32; the contour lines are drawn at intervals of 10m. are attempts to remove the distinct excitation by distributing the grid points more evenly over sphere, cf. (Tomita et al. 2001). 8.4 Test Case 5: Zonal Flow over an Isolated Mountain This case tests the ability of the scheme to conserve total energy when conversion between kinetic and potential energy takes place. It displays the full instationary shallow water dynamics and the initial data the geostrophic flow setting as in test case 2 rapidly generates a multitude of waves due to the presence of a single conic mountain in lower boundary. No analytical solution is known for this test case and the errors are computed with respect to a reference solution provided by Jakob et al. (1993). Two integral invariants of shallow water flow are monitored for this test case: total energy e = 1 2 hv v g((h + h s) 2 (h s ) 2 ), (22) and potential enstrophy ξ = 1 (ζ + f) 2. (23) 2 h Neither one is a tracer variable, i.e. they are not described by an equation in flux form. That makes it particularly interesting to see how the model can handle such conservation properties. For better detection of conservation violations Williamson et al. (1992) suggest the normalized integrals I[e(t)] I[e(0)] ite = I[e(0)] I[ξ(t)] I[ξ(0)] ipe = I[ξ(0)] and (24) (25) based on the global integral I[ ] to be graphed. Also the normalized integrals of the global invariants total energy and potential enstrophy have been computed and are visualized in Figure 9. The conservation properties with respect to these quantities are very good. Total energy and potential enstrophy both decrease, but at very slow rates. Several other models display the same or larger changes (Heinze 1998; Stuhne and Peltier 1999). Only a few seem to conserve energy better, (Thuburn 1997; Heikes and Randall 1995). Potential enstrophy also decreases but on scales smaller or comparable to other models, e.g. (Thuburn 1997). It is a natural effect that potential enstrophy will decrease in this test case as it is transferred to smaller scales, which are not resolved by the model. The development of the error norms during the model period is shown in Figure 10. Especially in the h-field the errors are mainly produced during the first day, afterwards they increase very slowly. The higher resolution improves the velocity field rather than the height field. The spatial distribution of the error in the height field is plotted in Figure Test Case 6: Rossby-Haurwitz Wave The sixth test case deals with a Rossby- Haurwitz wave, which is not a solution of the shallow water equations. In spite of this it has been used for a long time for meteorological tests and is included in the test set by Williamson et al. (1992). The reference solution again has been obtained with a high resolution spectral model. Compared with the reference solution the differences in the height field in Figure 12 are about one order of magnitude larger than in test case 5. This has also been found by other authors with grid point models and the results are of similar quality. 9 Summary An icosahedral grid-based FV discretization for the shallow-water equations has been developed. The icosahedral grid is an interesting approach

12 12 9 SUMMARY Figure 9: Normalized values for two integral invariants are shown for test case 5. Left: integrated total energy (ite, definition (22)), right: potential enstrophy (ipe, definition (23)), ILU- GMRES. Figure 10: Example of the l-norms of the differences between computed solution and reference solution for test case 5 (ILU- GMRES). for a smooth distribution of the grid points on the sphere and implies a data structure suitable for computationally efficient treatment. The prognostic equations were used in potential vorticity divergence form to describe the kinematic situation. The evolution of potential vorticity is described by a flux equation; consequently this is a second conservation property that can be respected by the treatment with a finite-volume scheme. This is achieved, however, at the cost of the inversion of the relationships that determine potential vorticity and divergence from the velocity components. It has been shown that the solution of the discrete Poisson equations is crucial for the algorithm. The most reliable method for solving the discrete system was a GMRES solver, whose performance varies depending on the preconditioner. The attempt to use an algebraic multigrid method was only partly successful; apparently the AMG preconditioner is too sensible in case of singular matrices and incompatible right hand sides. However AMG is always very effective and safe in the first one or two iterations and it might be possible to receive an increased performance by starting the solution procedure with one or two AMG-preconditioned steps after which a more stable method can be used if considered necessary based on the previous residual reduction. Especially at high resolutions where the standard methods converge more slowly this strategy might lead to a considerable reduction of the number of iterations that has to be performed. The elliptic solver is the component which possibly deserves most attention, and further effort in this field and a systematic comparison with the geometric multigrid methods used by other authors are certainly desirable for the future. The results that were presented are encouraging and justify further investigations. Conservation properties are of vital importance for atmospheric modeling and reliable compliance with these properties is achieved by the finitevolume discretization. The tests with the full shallow water dynamics showed good accordance with the reference solutions in comparison with other grid point methods. The small time steps that have to be taken because of the explicit scheme are a weak point: as the Adams- Bashforth scheme was implemented as an explicit method, the time step is limited by a

13 REFERENCES 13 Figure 11: Results for test case 5, ILU-GMRES, resolution ni = 32. The numerical solution for day 15 is on the left (grid spacing 100 m) and the difference to the reference solution on the right (grid spacing 5 m, solid lines: positive error, dashed lines: negative error). Figure 12: Results for test case 6. The height field is shown after the fourteenth day in the left column. The right column shows the difference between the computed solution and the reference solution. The spacing is 100 m for both figures. Positive errors are marked with solid lines, negative ones with dashed lines. CFL criterion containing the gravity wave speed of the system. Semi-implicit or implicit methods in connection with Lagrange methods can help to surmount this problem and allow much longer time steps. Although an implementation of these techniques would have been highly desirable it was not possible in the given time. A comparison of a linear implicit third-order Runge-Kutta-Rosenbrock method with an explicit Runge-Kutta method in the shallow water context is given in Lanser et al. (2000) such issues could be another field for further investigations with this model. References Baumgardner, J. R. (1983, November). A threedimensional finite element model for mantle convection. Ph. D. thesis, University of California, Los Angeles. Baumgardner, J. R. and P. O. Frederickson (1985, December). Icosahedral discretization of the two-sphere. SIAM J. Numer. Anal. 22 (6), Cassirer, K., W. Joppich, S. Pott, and R. Redler (2001). Ein Vorschlag für eine finite Differenzendiskretisierung von Laplaceund Diffusionsoperator auf dem Ikosaedergitter. Technical report, Institut für Algorithmen und wissenschaftliches Rechnen (SCAI), GMD Forschungszentrum Informationstechnik GmbH, Sankt Augustin. Durran, D. R. (1991). The third-order Adams- Bashforth method: An attractive alternative to leapfrog time differencing. Monthly Weather Review 119, Heikes, R. and D. A. Randall (1995). Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part I: Basic design and results of tests. Monthly Weather Review 123, Heinze, T. (1998, December). Ein numerisches Verfahren zur Lösung der Flachwassergleichungen auf einer rotierenden Kugel mittels der Lagrange-Galerkin-Methode. Diplomarbeit, Institut für Angewandte Mathematik der Rheinischen Friedrich-Wilhelms- Universität Bonn.

14 14 REFERENCES Jakob, R., J. J. Hack, and D. L. Williamson (1993). Solutions to the shallow water test set using the spectral transform method. NCAR Technical Note NCAR/TN-388+STR, NCAR. Jameson, A., W. Schmidt, and E. Turkel (1981). Numerical simulation of the Euler equations by finite volume methods using Runge-Kutta time stepping schemes. In AIAA 5th Computational Fluid Dynamics Conference. AIAA Paper Lanser, D., J. G. Blom, and J. G. Verwer (2000, July). Time integration of the shallow water equations in spherical geometry. Report MAS- R0021, CWI, Centrum voor Wiskunde en Informatica. Majewski, D. (1996). Documentation of the new global model (GME) of the DWD. Technical report, DWD. Pesch, L. (2002, December). A Finite-Volume Discretization of the Shallow-Water Equations in Sperical Geometry. Diplomarbeit, Meteorologisches Institut der Rheinischen Friedrich- Wilhelms-Universität Bonn. Stuhne, G. R. and W. R. Peltier (1999). New icosahedral grid-point discretizations of the shallow water equations on the sphere. J. Comp. Phys. 148, Thuburn, J. (1996). Multidimensional flux-limited advection schemes. J. Comput. Phys. 123, Thuburn, J. (1997, August). TVD schemes, positive schemes, and the universal limiter. Mon. Wea. Rev. 125, Tomita, H., M. Tsugawa, M. Satoh, and K. Goto (2001). Shallow water model on a modified icosahedral grid by using spring dynamics. J. Comput. Phys. 174, Williamson, D. L., J. B. Drake, J. J. Hack, R. Jakob, and P. N. Swarztrauber (1992). A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comp. Phys. 102,