Analysis of a mixed finite-element pair proposed for an atmospheric dynamical core

Authors

  • Andrew Staniforth,

    1. Met Office, Exeter, UK
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    • The contributions of these authors were written in the course of their employment at the Met Office, UK, and are published with the permission of the Controller of HMSO and the Queen's Printer for Scotland

  • Thomas Melvin,

    Corresponding author
    1. Met Office, Exeter, UK
    • Met Office, FitzRoy Road, Exeter EX1 3PB, UK.
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    • The contributions of these authors were written in the course of their employment at the Met Office, UK, and are published with the permission of the Controller of HMSO and the Queen's Printer for Scotland

  • Colin Cotter

    1. Department of Aeronautics, Imperial College London, UK
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Abstract

We present a numerical dispersion analysis for the P2 − P1DG finite-element pair applied to the linear shallow-water equations in one dimension. The aim is to provide insight into the numerical dispersion properties of the RT1 and BDFM1 finite-element pairs in two dimensions, which have recently been proposed for horizontal discretisations of atmospheric dynamical cores with quasi-uniform grids. This is achieved via analysis of a one-dimensional RT1 element. Whilst these finite-element pairs have been shown to have many desirable properties that extend properties of the C grid to non-orthogonal quadrilateral and triangular grids, including stationary geostrophic modes on the f plane, and a 2:1 ratio of velocity to pressure degrees of freedom (a necessary condition for the absence of spurious mode branches), it is also important to have appropriately physical numerical wave propagation. In the absence of Coriolis force, we compute the group velocity for P2 − P1DG. We find that, as well as dropping to zero at the grid-scale, which also occurs for the C-grid finite-difference method, the group velocity also drops to zero in a narrow band around kh = π which corresponds to eigenmodes with a wavelength close to two element widths. This is a potential problem because it increases the amount of wavenumber space that needs to be filtered. In this one-dimensional case, we find that this particular issue can be removed by a small modification of the equations, namely partially lumping the mass matrix, in such a way that the other favourable properties of the scheme are not affected. We discuss various symmetric and asymmetric modifications of the mass matrix, and show that both such modifications preserve energy conservation (having modified the definition of discrete kinetic energy). Finally we illustrate our findings with numerical experiments, and discuss the potential to extend this modification to two-dimensional schemes.

1. Introduction

In the context of the expected evolution of massively parallel supercomputer architectures, Staniforth and Thuburn (2012) have recently reviewed and assessed horizontal grids for modelling the atmosphere over the sphere. They concluded that all grids proposed to date have known problems or issues that merit further investigation. Traditional latitude–longitude-based grids lead to data communication bottlenecks due to resolution clustering around the two poles. However, although discretisations on alternative, quasi-uniform, grids reduce the severity of this problem, they are prone to other drawbacks such as computational modes and grid imprinting. Computational modes are potentially very serious, since they may be excited in realistic applications by boundary conditions, nonlinearity, physical forcing, and data assimilation.

Much of the current thrust of research on atmospheric dynamical cores is focused on discretisations that employ either a subdivided icosahedral grid, or a cubed-sphere grid (Lauritzen et al., 2011). In a recent development, Cotter and Shipton (2012) have proposed the use of some particular families of mixed finite elements as a means of improving the horizontal discretisation of atmospheric dynamical cores on such quasi-uniform grids. They demonstrate that these mixed elements preserve the following desirable properties of the C-grid method when applied to linear barotropic wave propagation:

  • 1.energy conservation;
  • 2.mass conservation;
  • 3.absence of spurious pressure modes;
  • 4.steady geostrophic modes on an f-plane; and
  • 5.the flexibility to adjust the balance of velocity degrees of freedom to pressure degrees of freedom, so as to achieve a 2:1 ratio.

Regarding the last property, it was shown by Cotter and Shipton (2012) that the 2:1 ratio is a necessary condition for absence of spurious mode branches for this family of mixed finite-element methods.

Two examples of mixed finite elements are specifically recommended: the recently developed modified Raviart–Thomas (RT) element on quadrilaterals, and the Brezzi–Douglas–Fortin–Marini (BDFM) element on triangles. The lowest-degree RT finite-element space is the mixed finite-element analogue of the C grid: the pressure space has piecewise constant (discontinuous) functions, and the velocity fields are bilinear, but constrained to have constant, continuous normal components on element edges. This mixed element shows promise for discretisation on a cubed sphere.

The second of Cotter and Shipton (2012)'s proposed mixed elements is based on triangles, and offers promise for discretisations on triangularly subdivided icosahedral grids. The lowest possible order for triangular elements that allows a 2:1 ratio is the BDFM1 element pair (piecewise linear and discontinuous for pressure, piecewise quadratic with continuous normal components for velocity, with the constraint that normal components are linear on edges). In contrast to the quadrilateral case, where the lowest possible order is RT0, which is built from constant and linear functions, the velocity representation for BDFM1 contains quadratic functions (although not all of them). Therefore a key question is whether or not the inclusion of quadratic functions for these triangular elements is likely to adversely affect numerical dispersion: a good representation of group velocity is of crucial importance since it controls energy propagation. In this article, we take steps towards understanding the dispersion properties of BDFM1 by studying the RT1 space on quadrilaterals, which is the quadrilateral analogue of BDFM1 and also has quadratic functions in the velocity expansion. If the RT1 space on quadrilaterals turns out to suffer from poor dispersion properties, then it seems likely that BDFM1 will also inherit the same poor dispersion properties. An important advantage of this approach is that, for the rectangular case, the problem can, under the assumption of variation in only one direction, be reduced to a one-dimensional (1D) problem, thereby greatly facilititating analysis tractability.

A mixture of discontinuous and continuous finite-element spaces has long been employed for fluid dynamical problems, as one way of satisfying the Babuska–Brezzi stability condition. The P2 − P1DG scheme considered here in one dimension can be extended to several different schemes in two dimensions depending on the continuity and extra degrees of freedom in the extra direction. The most obvious extension is to take Q2 − Q1DG on quadrilaterals in two dimensions, which forms part of the Q(k) − Q(k − 1)DG high-order family that was proposed for ocean modelling by Iskandarani et al. (1995). In that article, a spectral element approach was used in which incomplete (but consistent) quadrature leads to a diagonal mass matrix. In the context of engineering flows, the related P(k) − P(k − 2)DG space on triangles was introduced for the Navier–Stokes equations by Deville et al. (2002). A fully discontinuous QkDGQkDG approach was introduced by Giraldo et al. (2002), and extended to PkDGPkDG triangles by Giraldo (2006). The quest to find schemes without spurious modes has led to the consideration of partially continuous spaces for velocity, including the Brezzi–Douglas–Marini (BDM), the BDFM, and RT spaces, which are all described by Brezzi and Fortin (1991). The dual scheme P1DGP2 was proposed by Cotter et al. (2009a). The dispersion properties of a number of these schemes have been analysed, including equal-order DG schemes (Ainsworth et al., 2006), RT and BDM schemes (Rostand and Le Roux, 2008), and P1DGP2 by Cotter and Ham (2011) and Le Roux (2012). Dispersion relations for a number of other finite-element spaces were analysed by Le Roux et al. (2007) and Le Roux et al. (2008). A number of spaces were analysed by inspecting the matrix structure in Le Roux et al. (2005), and five discontinuous and partially discontinuous mixed finite-element spaces were investigated numerically by Comblen et al. (2010). The approach of modifying the mass matrix to improve dispersion properties was investigated by Gassner and Kopriva (2011), although the types of lumping were quadrature-based unlike the partial lumping discussed in this article.

The continuous 1D linear shallow-water equations used in this study are given in section 2. The mixed-element discretisations of these equations are derived in appendices and summarised in section 3. A dispersion analysis of a subset of these equations, the gravity wave equations, is presented in section 4. This reveals a dispersion problem for waves with wavelengths equal or close to twice the element width. A means of addressing this issue for the considered 1D problem is described in section 5. Illustrative numerical integrations of the 1D gravity wave equations are presented in section 6. Energy conservation for the proposed solution is then examined in section 7. Finally, conclusions are drawn in section 8, including a brief discussion of outstanding issues.

2. 1D linear equation sets

In this section we establish notation for the rotating shallow-water equations and recall the dispersion properties of gravity wave propagation when f is identically equal to zero.

2.1. Shallow water

The 1D linear shallow-water equations on a f-plane are

equation image(1)
equation image(2)
equation image(3)

where u(x,t) and v(x,t) are the velocity components in the x and y directions, respectively, Φ(x,t) ≡ gH(x,t), H(x,t) is the height of the fluid above a level surface, f is the constant Coriolis parameter of the f-plane, and g is constant acceleration due to gravity. By analogy with the finite-element literature, Φ(x,t) plays the role of pressure. Equations (1)–(3) are used to examine energy conservation of the original and modified Cotter and Shipton (2012) discretisations.

2.2. Gravity wave

The subset of 1D linear gravity wave equations is adopted for the dispersion analysis of section 4. Setting f identically zero in (1)–(3) gives

equation image(4)
equation image(5)

where the v momentum equation has been dropped since it decouples from (4)–(5). In preparation for the dispersion analysis, assuming harmonic solutions of the form exp[i(kxωt)] in (4)–(5) leads to

equation image(6)

where ωexact, cexact and cgroup are the exact frequency, phase velocity and group velocity, respectively.

3. Mixed element discretisation

3.1. Mixed element representation

The analogue on a rectangular grid of Cotter and Shipton (2012)'s proposed mixed element on a triangular grid (their Figure 1b) is displayed in Figure 1, and has the following characteristics.

Figure 1.

Schematic for the placement of variables for mixed linear-quadratic elements on a rectangle: normal components of velocity held at filled circles, tangential components at open circles, and Φ (pressure) at open squares.

The pressure field is represented as a bilinear function within each element, with no cross-element continuity. It is best to interpret this discontinuous representation as a projection of the continuous pressure field onto bilinear functions within each element. This should be compared with the C-grid representation of pressure, which is regarded as the cell average of the pressure, i.e. the projection of the continuous pressure field onto constant functions within each element. Choosing the Lagrange basis for bilinear functions in an element with nodal points at the element corners results in basis coefficients taking values of the bilinear functions in the corners, with a different value for each element surrounding the corner.

The velocity field is represented in each element as a biquadratic vector-valued function with the constraint that, when restricted to each edge, the edge-normal component of the function is linear. The velocity field representation has continuous normal components across element boundaries, but the tangential components are allowed to jump across element boundaries. This is enough to allow the divergence of the velocity field to be computed, but not the curl: such representations are called div-conforming. Again, it is best to interpret these partial discontinuities as projections. For the C grid (and RT0), the normal components of velocity that are stored on edges can be interpreted as edge-averages of the normal components of the continuous velocity. For RT1, the linear edge-normal components are projections of the continuous field onto linear functions on the edge. The interior degrees of freedom are obtained from additional projections of the continuous field onto linear functions within each element, plus the unique quadratic incompressible vector field that vanishes on the element boundary.

Consider now the special case when it just so happens that the pressure and velocity fields to be represented have no y variation, only x variation. If this is true at initial time for the 1D gravity wave problem, then it will be true for all time, since there is nothing in the discretisation that would introduce any y variation. For this special situation, there just happens to be cross-element continuity in the y direction for both the pressure and the tangential component of velocity. Taking all this into consideration, it is seen that this mixed element on a rectangular grid is equivalent to using a mixed 1D element (Figure 2) with the following characteristics.

Figure 2.

Schematic for mixed linear-quadratic elements in 1D: u held at closed circles, v at open circles, and Φ (pressure) at open squares. Left and right element boundaries are denoted by tick marks.

Pressure varies linearly within an element, with no cross-element continuity: pressure is held at interval boundary points, with two, discontinuous, values associated with each element boundary, one to the immediate left, the other to the immediate right. There is only one component of velocity, u, and it varies quadratically within an element. It is held at interval boundaries, and also at interval midpoints. There is cross-element continuity of u, but not of its x derivative. Note that, at interval boundaries, u actually represents the normal component of velocity there of the 2D rectangular mixed element, from which this equivalent 1D mixed element was constructed. (Note that for the 1D gravity wave equations examined herein, the Cotter et al. (2009b) and Cotter and Shipton (2012) mixed elements lead to discrete equation sets that are formally equivalent, with the roles of the dependent variables u and Φ being interchanged. This equivalence does not however hold in two dimensions.)

The piecewise representations for these 1D mixed elements are given explicitly below. If one were to explicitly derive the discrete equations using the mixed quadrilateral element on a rectangular grid then, under the assumption of no y variation in the fields initially, these discrete equations would be the same along any line varying in x as those obtained using the 1D elements employed herein.

3.1.1. Piecewise continuous quadratic representation of u

u(x,t) is expanded in terms of piecewise-quadratic functions (Figure 3) that are continuous across element boundaries x = xm, so that

equation image(7)

where

equation image(8)
equation image(9)
Figure 3.

Quadratic basis functions over the interval [xm, xm+1]: Nm+1/2(x) (solid line), and Nm(x) and Nm+1(x) (dashed and dash-dot lines respectively).

3.1.2. Piecewise discontinuous linear representation of v

v(x,t) is expanded in terms of piecewise linear functions (Figure 4) that are discontinuous across element boundaries, so that

equation image(10)

where

equation image(11)
equation image(12)

equation image is the discontinuous value of v just to the left of x = xm, and equation image just to the right.

Figure 4.

Linear basis functions over the interval [xm, xm+1]: equation image (solid line), and equation image (dotted line).

3.1.3. Piecewise discontinuous linear representation of Φ

Φ(x,t) is expanded in terms of piecewise linear functions that are discontinuous across element boundaries, so that

equation image(13)

3.2. Discretisation of the 1D shallow-water equations

Various integrals need to be evaluated for the mixed-element discretisation of the 1D shallow-water equations. These are obtained by appropriate use of the numerical quadrature formulae given in Appendix A. Simpson's rule is used when the polynomial to be integrated is of cubic degree or less, and Bode's rule when instead it is of quartic degree. Because these rules are exact for these polynomials, this is equivalent to analytical integration. (Integrals could alternatively be evaluated using Gaussian quadrature of sufficient order, or exactly by hand, but this would still result in the same set of discrete equations.)

The discretisations of the 1D shallow-water equations (1)–(3) are derived in Appendices B–D. These are obtained by inserting the mixed-element representations (7) and (13) into the weak form of (1)–(3), and eliminating the integrals.

From Appendix B, the two independent discretisations (B4) and (B6) of the u momentum equation after normalisation are

equation image(14)

which is centred on an interval midpoint x = xm+1/2, and

equation image(15)

which is centred on an element boundary x = xm. A uniform mesh has been assumed such that hxm+1xm for all m, where {xm} is the set of points that define element boundaries.

Similarly, from Appendices C–D, the independent discretisations of the v momentum and continuity equations after normalisation are

equation image(16)
equation image(17)
equation image(18)
equation image(19)

4. Dispersion analysis of the 1D gravity-wave equations

The appropriate choice of equation set is the simplest one sufficient to examine the issue under consideration. The essence of the numerical dispersion issue for the mixed-order finite elements considered herein is embodied in the discretisation of the 1D gravity wave equations (4)–(5). The discrete counterparts of these two equations are (14)–(15) with f set identically zero, i.e.

equation image(20)
equation image(21)

together with (18)–(19). For analysis purposes, solving (18)–(19) for equation image and equation image allows them to be equivalently rewritten as

equation image(22)
equation image(23)

Equations (20)–(23) then comprise, for integer m, a set of four independent prototypical coupled equations for the four unknown prototypical quantities um, um+1/2, equation image and equation image. The numerical dispersion properties of this coupled equation set can be determined by decoupling um and um+1/2 from equation image and equation image under the assumption of exact representation in time.

Using (22)–(23) and their index decrements, equation image, equation image, equation image and equation image may be eliminated from (20)–(21) to obtain

equation image(24)
equation image(25)

Equations (24)–(25) are two coupled centred equations for u at half-integer and integer locations, to be solved over an assumed periodic domain. The same technique as that employed in Melvin (2012) to solve a similar set of coupled discrete equations is used here.

Therefore, seeking solutions of the form

equation image(26)

and inserting these into (24)–(25) gives (27).

equation image(27)

Using the identity cos(kh) ≡ 2cos2 (kh) − 1, and taking the determinant of the matrix in (27), then leads to the numerical dispersion relation

equation image(28)

Equation (28) is a quartic in ω, and a quadratic in ω2. It is therefore straightforward to solve it explicitly to obtain

equation image(29)

where

equation image(30)

(The first forms of (29)–(30) are particularly suited to examining behaviour for kh close to zero and 2π, and the second forms for kh close to π.)

After use of trigonometric identities, it can be shown that (29) is equivalent to the dispersion relation given on p 342 of Cotter et al. (2009b). The normalised (numerical divided by exact) frequency is displayed in Figure 5(a) for the positive ω branch as a function of non-dimensional wavenumber kh where, from (6), the exact frequency is equation image. This is equivalent to Figure 2 of Cotter et al. (2009b), with their top curve ‘folded over’ and replotted with πkh ≤ 2π. There is a spectral gap at kh = π, i.e. on the scale of the element width h. Although this gap looks innocuous, it is not.

Figure 5.

Numerical dispersion for the 1D linear-quadratic mixed elements (solid) of Cotter et al. (2009a, 2009b) and Cotter and Shipton (2012); staggered second-order finite differences on a C grid (dashed) at double resolution, and exact (dotted). (a) normalised (numerical divided by exact) frequency ω, and (b) normalised (numerical divided by exact) phase (black) and group (red) velocities.

The normalised (numerical divided by exact, with exact given in (6)) phase and group velocities cphaseω/k and cgroupdω/dk that correspond to (29) are plotted in Figure 5(b) for the positive ω branch. (Recall that this corresponds to applying 1D linear-quadratic mixed elements to the equations in the forms proposed in Cotter et al. (2009a, 2009b) and Cotter and Shipton (2012).) For comparison purposes, the corresponding results (dashed) for staggered second-order finite differences on a C grid at double resolution (they have the same number of degrees of freedom as the linear-quadratic mixed elements) are also shown.

For the C grid at double resolution, the group velocity drops to zero at kh = 2π, as is well known. This also occurs for the linear-quadratic mixed element, but there is an additional problem: the group velocity also drops to zero for wavelengths at and near kh = π, i.e. at wavelengths equal or close to twice the element width. For increasing kh in this narrow band of wavelengths, the normalised group velocity rapidly, and anomalously, goes from order unity (its exact value) to zero, and then back again to order unity. Because group velocity crucially controls energy propagation, this analysis suggests that the gravity wave energy associated with this band of wavelengths is likely to have singular propagation properties, with probable spurious local trapping of gravity wave energy. Thus, as formulated, the 1D linear-quadratic mixed element is unlikely to be a viable element. The question now is whether anything can be done to remove the singular spectral gap without adversely affecting the desirable properties of Cotter and Shipton (2012)'s mixed finite-element schemes.

5. Modified discretisation of the 1D gravity wave equations

In this section we attempt to modify the discretisation to remove the spectral gap whilst preserving the other properties of the method. Since one of the key properties is the steady geostrophic states on the f-plane, we concentrate on altering the time derivative term, so that the steady-state solutions are not changed. Hence, we restrict ourselves to modifying the mass matrix.

5.1. Partial mass lumping

Consider a modification of the time tendency terms in the momentum equations such that (20)–(21) are rewritten as

equation image(31)
equation image(32)

where α, β and γ are three parameters to be determined. In finite-element parlance, this approach corresponds to a partial mass-lumping of the time-tendency terms. (Although one could consider also introducing partial mass lumping of the time tendency terms in the continuity discretisations (18)–(19), this is not done here both to keep things simple, and because it does not appear to be necessary or beneficial.)

5.2. Dispersion analysis

The modified discretisations (31)–(32) lead to (24)–(25) mutating into

equation image(33)
equation image(34)

Seeking solutions of the form (26) and inserting these into (33) and (34) then leads to (35).

equation image(35)

Taking the determinant of the matrix in (35) gives the numerical dispersion relation

equation image(36)

To examine what happens at the problematic scale, set kh = π in (36) to obtain

equation image(37)

To remove the discontinuity in ω at kh = π implies that there be only one pair of roots, rather than two pairs. From (37), this can only happen if

equation image(38)

Thus (38) is the condition for removing the jump discontinuity in ω at the problematic scale kh = π. It constrains the three parameters α, β and γ, so there are in reality only two free parameters, rather than three. The constraint (38) suggests that γ = 1/10, which corresponds to the unmodified discretisation (20), is a particularly good choice. With this value, equation image from (38), which agrees with the exact result to less than one-tenth of a percent. It is equal to the smaller of the two values of the original discretisation (equation image), and better than its larger value equation image.

Using (38) to eliminate α, (36) reduces to

equation image(39)

Equation (39) is a quartic in ω, and a quadratic in ω2. It is therefore straightforward to solve it explicitly to obtain

equation image(40)

where

equation image(41)

It turns out that, because of the factor cos(kh/2) in front of equation image in (40), which was extracted from the square root, and the fact that cos(kh/2) changes sign at kh = π, the appropriate expression to plot for the dispersion relation is simply (40) with the positive choice of signs in front of square roots, and this is valid over 0 ≤ kh ≤ 2π.

5.3. Parameter setting

It remains to choose appropriate values for the three parameters (α,β,γ), of which only two are free since they must collectively satisfy the constraint (38) in order to remove the jump discontinuity in ω at the problematic scale kh = π. Two approaches were tried for this.

5.3.1. Symmetry of the partially lumped local mass matrix

The first involves what is known in the finite-element literature as the local mass matrix. For the partial mass lumping defined by (31)–(32), the associated local mass matrix is

equation image(42)

In the finite-element literature it is usual, as a means of ensuring energy conservation, to retain symmetry of the local mass matrix ℳe when mass lumping. It is therefore natural to set 2γ = β so that the partially lumped local mass matrix (42) is then symmetric. Using this value in the constraint equation (38) then allows α and γ to be expressed in terms of the free parameter β as

equation image(43)

Varying β, it was found that setting β = 1/5 (which corresponds to α = −3/20 and γ = 1/10) leads to good dispersion properties, which are displayed in Figure 6 for this choice. Comparison of this figure with the corresponding one (Figure 5) for the unmodified scheme shows that the anomalous behaviour has been surgically removed, as desired. By construction, there is no longer a spectral gap at khπ for the normalised frequency (Figure 6(a)); the phase and group velocities both also behave well at and around kh = π, and both are quite accurate for most of the spectrum (Figure 6(b)).

Figure 6.

As Figure 5, but for the modified scheme with (α,β,γ) = (−3/20,1/5,1/10).

5.3.2. Setting γ ≡ 1/10

The second approach tried was to set γ ≡ 1/10 which, as argued above, leads to a highly accurate value for ω|kh=π. Using this value in the constraint equation (38) then leads to

equation image(44)

Varying β, it was found that setting β ≡ 0 (which corresponds to α = −1/20 and γ = 1/10) leads to excellent numerical dispersion properties, which are displayed in Figure 7 for this particular choice. By construction, there is again no longer a spectral gap at khπ for the normalised frequency (Figure 7(a)); the phase and group velocities both also behave very well at and around kh = π, and both are highly accurate for most of the spectrum (Figure 7(b)).

Figure 7.

As Figure 5, but for the modified scheme with (α,β,γ) = (−1/20,0,1/10).

6. Numerical integrations of the 1D gravity-wave equations

To practically illustrate the identified dispersion problem for wavelengths equal or close to twice the element width, various time integrations of the 1D gravity-wave equations are performed using the mixed-element discretisation, without and with the proposed modification. For comparison purposes, integrations are also performed using second-order finite differences on a staggered Arakawa C grid having the same number of degrees of freedom as for the mixed-element discretisation, i.e. the meshlength is half the element width. All time integrations are performed to time t = 0.75 using Matlab software with Dormand and Prince (1980)'s explicit (4,5) Runge–Kutta timestepping scheme, with 120 elements in a two-units-wide periodic domain, and with Φ0 set to unity.

The initial structure for Φ is chosen to correspond to a pure kh = π mode, modulated by a Gaussian envelope equation image of unit amplitude, where a = 0.1 is its half-width parameter (Figure 8). The initial wind field u(x,t = 0) is set identically zero. Thus

equation image(45)
equation image(46)

where xmmh. These initial conditions (Figure 8) correspond to a pair of leftward and rightward, symmetrically propagating, gravity waves, each having half-unit amplitude, which reinforce one another at initial time to give unit amplitude. The exact solution after time t = 0.75 is comprised of two symmetrically located wave packets of half-unit amplitude: these are centred on x = ±0.75 since the exact group velocity is equation image.

Figure 8.

Results using the unmodified mixed-element scheme; this corresponds to (α,β,γ) = (−1/10,1/5,1/10). (a) u and (b) Φ at time t = 0.75, with the initial condition for Φ plotted in blue, and time evolution of (c) u and (d) Φ as a function of x. Contour interval is 0.05 for all panels, and only positive contours are plotted on (c) and (d).

Results of the integrations using: the unmodified mixed-element scheme; the 2nd-order finite difference scheme on a C grid; and the modified mixed-element scheme with (α,β,γ) first set to (−3/20,1/5,1/10), and then to (−1/20,0,1/10); are displayed in Figures 8–11, respectively. It is seen (Figure 8) that the unmodified scheme behaves in an unphysical manner, as predicted by the analysis of section 4. Although there are two symmetric wave packets approximately centred about their correct locations, they are of reduced amplitude. Furthermore, and importantly, there is significant spurious wave activity between them, of similar amplitude, where there should be none. This illustrates the poor dispersion for disturbances with wavelengths equal or close to twice the element width, as uncovered by the analysis of section 4.

Figure 9.

As Figure 8, but for the C-grid scheme at double resolution.

Figure 10.

As Figure 8, but for the modified mixed-element scheme with (α,β,γ) = (−3/20,1/5,1/10). This set of parameters leads to a symmetric mass matrix.

Figure 11.

As Figure 8, but for the modified mixed-element scheme with (α,β,γ) = (−1/20,0,1/10). This set of parameters leads to an asymmetric mass matrix.

The C-grid scheme behaves in a much more physical manner (Figure 9). The two wave packets propagate in a properly separated manner, albeit too slowly. This is however entirely consistent with analysis which predicts (Figure 5(b)) that the group velocity is retarded by approximately 30% for wavelengths around kh = π, and this retardation is indeed evident in Figure 9. The modified scheme of section 5.3.1, with (α,β,γ) set to (−3/20,1/5,1/10), also behaves in a physical manner. However, whereas the group velocity is retarded by the C-grid scheme by about 30%, the modified scheme of section 5.3.1 accelerates it by approximately 15%: this is consistent with the value at kh = π in Figure 6(b). The best-performing scheme, again consistent with analysis, is the modified mixed-element scheme with (α,β,γ) set to (−1/20,0,1/10) (Fig. 11). There is virtually no retardation (consistent with Figure 7(b)), and the half-unit amplitude for Φ is much better maintained than any of the other three schemes.

7. Energy conservation for the 1D shallow-water equations

In the context of the 1D gravity-wave equations, the modified discretisation proposed in section 5 has been shown in section 6 to address the dispersion problem identified by the analysis of section 4. One of the desirable features of the Cotter and Shipton (2012) schemes is energy conservation. The question thus arises as to whether the modified discretisation still conserves energy, not only for the 1D gravity-wave equations, but also for the 1D shallow-water equations.

Consider therefore the following set of discrete approximations to the 1D shallow-water equations (1)–(3):

equation image(47)
equation image(48)
equation image(49)
equation image(50)
equation image(51)
equation image(52)

Introducing partial mass lumping into the discrete analogues (14)–(15) of the u−momentum equation, cf. (31)–(32), leads to (47)–(48). Equations (49)–(50) are the result of having introduced a partial mass lumping into the discrete analogues (16)–(17) of the v−momentum equation, such that setting δ = 0 recovers them. Equations (51)–(52) are just a repetition of (18)–(19), included here for readability.

At this point there are four mass-lumping parameters, viz. α, β, γ and δ, but only two of them are independent. The parameter α = α(β,γ) is already constrained to satisfy (38) in order to remedy the spurious dispersion problem. It then turns out –see (60) –that δ is constrained to be a particular function of β and γ in order to conserve energy.

Taking equation image gives

equation image(53)

When Fm+1/2 = 1, (47) is recovered. When Fm+1/2 = um+1/2, the left-hand side contributes to the time rate of change of kinetic energy for the interval [xm,xm+1], and the right-hand side to cancelling part of the rate of change of potential energy and to cancelling the Coriolis terms. Rewrite (48) as

equation image(54)

The top two lines originate from the interval [xm,xm+1], whereas the lower two lines originate from the interval [xm−1,xm]. Focusing now on the interval-by-interval contributions of (54) to the kinetic energy budget, consider just the interval [xm,xm+1]. For this interval, there are contributions from the first three terms of (54), viz.

equation image(55)

plus a further three terms

equation image(56)

which are obtained by incrementing the index of the second line of (54). Now take [Fm×(55) + Fm+1×(56) ]/3, and set the sum over all intervals to zero to obtain

equation image(57)

Setting Fm = δkm in (57), where δkm here is exceptionally the Kronecker delta function, recovers (54), but with m replaced by k. Thus (48) is recovered, since (54) is simply (48) rewritten, and so (57) contains (48) as a special case.

Now set Fm = (1 − δ)um + δum+1/2 and Fm+1 = (1 − δ)um+1 + δum+1/2 in (57) to obtain (58).

equation image(58)
equation image(59)

Setting Fm+1/2 to um+1/2 in (58), summing the result over m, and adding this to (59) then leads to (59).

By choosing

equation image(60)

the terms in the second line of (59) can be combined into a single time derivative, yielding (61).

equation image(61)

The contribution to the kinetic energy from the v−momentum equation can be obtained by taking [equation image(49) + equation image (50)]/2, and summing over all elements, so that

equation image(62)

Taking Φ0× [(61) + (62)], the fluxes from the Coriolis terms cancel and the resulting kinetic energy budget equation is

equation image(63)

where equation image is the appropriate measure of kinetic energy in the interval [xm,xm+1], given by (64).

equation image(64)

Taking [ equation image (51) + equation image (52)] (1 − δ)/2 gives the potential energy budget equation (65), where

equation image(65)
equation image(66)

is the appropriate measure of the potential energy in the interval [xm,xm+1]. For energy conservation, the right-hand side of (65) must, when summed over the domain, cancel similar summed contributions from the rate of change of total kinetic energy. Thus summing (65) over m gives

equation image(67)

and summing (67) and (64) then leads to

equation image(68)

i.e. to energy conservation.

As a cross check, when

equation image(69)

(64) and (66) reduce to

equation image(70)

as they should.

The metric (66) for measuring potential energy is independent of the mass-lumping parameters, and unchanged from the original Cotter and Shipton (2012) formulation. However the metric (64) for measuring kinetic energy does depend upon the mass-lumping parameters. Thus, although total energy is conserved by the modified discretisations, it is conserved with respect to a subtly-different measure of kinetic energy than that appropriate for the original Cotter and Shipton (2012) formulation.

8. Conclusion

It has been shown that the Cotter and Shipton (2012) formulation, when applied to the 1D gravity-wave equations, leads to a dispersion problem for wavelengths equal or close to twice the element width. This has serious implications for energy propagation since this propagates with the group velocity, which spuriously and strongly goes to zero for waves with wavelengths equal or close to twice the element width. It also strongly suggests that 2D quadrilateral and triangular mixed elements are likely to prove problematic.

However, it is demonstrated that the dispersion deficiency can, for this model problem, be addressed by a partial lumping of the mass matrix associated with the time-tendency term of the momentum equation.

Having done so in the context of the 1D gravity-wave equations, the approach was then extended to the discretisation of the 1D shallow-water equations. Since one of the desirable features of the Cotter and Shipton (2012) schemes is energy conservation, the question thus arises as to whether this property is retained by the proposed partial mass-lumped discretisations. This was found to be the case, provided that an appropriate discretisation of the v momentum equation is adopted, and that an appropriate measure for kinetic energy is used.

It is an open question as to whether the partial mass lumping approach can be successfully extended to 2D discretisations with quadrilateral and triangular elements. This is currently being investigated.

Acknowledgement

The authors thank Dr Markus Gross for his assistance in using the Maple software package.

Appendix A. Numerical quadrature formulae

A.1 Simpson's rule

Simpson's rule applied over a single interval is

equation image(A1)

where hxm+1xm. This quadrature is exact for cubics.

A.2 Bode's rule

Bode's rule applied over a single interval is

equation image(A2)

which is exact for quintics.

Appendix B. Discretisation of the u momentum equation

Cotter and Shipton (2012) discretise the momentum equation by first writing it in the weak form

equation image(B1)

with test functions w, which are then restricted to the P2 finite element space, wN(x) ∈ P2, giving

equation image(B2)

B.1 Orthogonalising the error to equation image

Setting equation image into (B2) and using variable expansions gives (B3).

Elimination of the integrals then yields

equation image(B3)
equation image(B4)

where a uniform mesh has been assumed such that hxm+1xm for all m, and {xm} is the set of points that define element boundaries.

B.2 Orthogonalising the error to Nm (x)

Similarly expanding, but setting N(x) = Nm (x), which is non-zero over two intervals rather than the single interval of the above cases, gives (B5). Eliminating the integrals then yields

equation image(B5)
equation image(B6)

Appendix C. Discretisation of the v momentum equation

The v momentum equation is written in the weak form

equation image(C1)

and the test functions ϕ are then restricted to the P2 finite element space, ϕM(x) ∈ P2, so that

equation image(C2)

C.1 Orthogonalising the error to equation image

Inserting variable expansions into the continuity equation (5), and orthogonalising the error to a basis function equation image, gives (C3).

Elimination of the integrals then yields

equation image(C3)
equation image(C4)

C.2 Orthogonalising the error to equation image

Similarly expanding, but orthogonalising the error to equation image, gives

equation image(C5)

Appendix D. Discretisation of the continuity equation

D.1 Orthogonalising the error to equation image

Inserting variable expansions into the continuity equation (5), and orthogonalising the error to a basis function equation image, gives (D1). Elimination of the integrals then yields

equation image(D1)
equation image(D2)

D.2 Orthogonalising the error to equation image

Similarly expanding, but orthogonalising the error to equation image, gives

equation image(D3)

Ancillary