Open boundary conditions for horizontal 2-D curvilinear-grid long-wave dynamics of a strait. A. A. Androsov,= K. A. Klevanny,b E. S. Salusti' & N. E. Vohingerâ.
Advances in Water Resources Vol. 18, No. 5, pp. 267-276, 1995 Copyright 0 0309-1708(95)00017-8
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1995 Elsevier Science Limited
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Open boundary conditions for horizontal 2-D curvilinear-grid long-wave dynamics of a strait A. A. Androsov,= K. A. Klevanny,b E. S. Salusti’ & N. E. Vohinger” “St Petersburg Branch, Institute of Oceanology, Russian Academy of Sciences, 30, Pervaya Liniya, 199053, St Petersburg, Russia bState Hydrological Institute, 23, Vtoraya Liniya, 199053, St Petersburg, Russia ‘INFN, Dipartimento di Fisica, Universita delgi Studi di Roma ‘La Sapienza’, P. le A. More, 2 00185 Rome, Italy
(Received 16 May 1994; accepted 8 May 1995)
Shallow-water equations in a domain representing a strait are transformed into curvilinear coordinates concordant with the domain configuration and mapping the domain onto a rectangle with two opposite open boundaries. The initial boundary value problem for the equations in the form of contravariant fluxes is considered and compared with a reduced formulation of boundary conditions for equations linearized near the boundary. A method for numerical integration of the transformed boundary value problem is discussed. Particular attention is given to modification ensuring stable computation in the case of strong nonlinearity and variability of domain metric. Kcv words: boundarv-value curvilinear coordinates.
nroblem,
1 INTRODUCTION
correct
and
reduced
formulations,
favorable forms being excluded. To set the correct boundary conditions for the non-linear shallow water equations it is necessary to have information not only about the water level, but also about the velocity field, while the latter is generally not feasible. As a result, one must confine oneself to the cases where it is enough to set the water level at the open boundary, and to equate tangential velocity at the inflow to zero, if possible. Another simplified method is to use a cutoff function whereby advection is not computed at the boundary and the second boundary condition is not set at the inflow. As will be shown this reduced formulation is justified and is appropriate only if the boundary lies in the deep zone where advection is negligible. In hydrodynamical modeling based on numerical integration of the quasilinear shallow-water equations the statement relating to correct boundary conditions, with exception (Elvius & Sundstr6m,4 Sivashinsky13), is not fulfilled. This circumstance is a motivation of emphasized attention to topic. In the case of complex shoreline geometry, it is useful to transfer to the curvilinear coordinates concordant with the domain configuration; the problem can thus be considered in a canonical domain, for example, in a rectangle (Johnsson Haeuser et al.,’ Sheng,12 Spaulding,14 Voltzinger and Klevanny’*). In the computational counterpart of a physical domain representing a strait the computational domain is a rectangle with two opposite open boundaries. Boundary conditions at the
Numerical integration of the quasi-linear hyperbolic shallow-water equationis in a domain representing a strait encounters two peculiarities which demand our attention. The first results from the necessity for a satisfactory description of a real, usually rather complex, coastal configuration. A widely used approach is to approximate the bound,ary by segments parallel to the coordinate axis. This leads to distortion of the solution in the coastal zone which cannot be removed by detailing the grid (Goto & Shuto2). Secondly, setting boundary conditions at open boundaries involves some difficulties. The permissible boundary conditions have been analyzed by Oliger and Sundstriim.” A fruitful approach used in meteorology is to telescope a given domain and to set the required conditions at its boundaries obtaining them by preliminary computation on a coarse grid in the extended region (Sundstriim & Elvius15). However, in this case its applicability is restricted, because in order to obtain the solution of oceanology boundary value problems in an expanded domain, provided that this expansion is possible, to be no simpler than for a telescoped domain. It is necessary to add thiat, in practice, the most readily available information at the open boundary is the water level, and this significantly restricts the class of admissible boundary conditions, the most computationally 267
A. A. Androsov et al.
268
sides of the rectangle are set according to the demands of the correct formulation of the problem for a quasilinear hyperbolic system (Kreiss’), but these demands for the transformed equations have a specific character. This paper first of all deals with the boundary value problem for shallow-water equations in the form of contravariant fluxes and also with a numerical method for integration of the transformed problem. The next section describes transfer to the boundary value problem for contravariant fluxes. The conditions on the characteristics define an invariant which includes the metric of transformation and represents the relation between the level and the lengthwise contravariant component of velocity. Another characteristic variable, the tangential velocity, is transformed into a transversal covariant component of velocity. Section 3 presents a numerical method for integration of the transformed problem on the rectangular grid which is the map of the corresponding grid of a physical domain. The equations based on the semi-implicit method are approximated by the scheme of Crank-Nicholson implemented by splitting. In spite of the fact that method has been previously tested and widely applied (Klevanny et aZ.,7 Voltzinger et a1.19)some modifications turn out to be necessary to adapt it to this problem. The main change relates to the solution of the equation for the transversal component of velocity at the first half-step since it is especially susceptible to the high-frequency disturbances of the solution. Therefore it was replaced by an equation for the corresponding covariant component. Section 4 contrasts both methods of setting conditions at an open boundary. In situations where the transversal covariant component of velocity near the boundary is comparable with its lengthwise contravariant component, the reduced formulation of the problem becomes very dubious. This result has been verified by comparative computation of the M2-wave dynamics in the Strait of Messina for problems in correct and reduced formulation. It is worth noting that the Strait of Messina is a highly suitable example for clearing up a distinction in these two formulations because of the strong variability of morphometric data, high nonlinearity and intensity of tidal dynamics caused by out-of-phase level oscillations M2-wave at boundaries (Bignami & Salusti’). A comparison was fulfilled by computation of the M2wave total energy in a tidal cycle for each considered problem. Special attention was paid to fulfilling the energy balance of the physical factors in the simulated process. Equations of energy in curvilinear coordinates and its difference analog are presented in the Appendix.
2 FORMULATION
OF THE PROBLEM
value problem
u, + Au, + BUY= ti,
for the
X,Y E fl2,
t10
(1)
u u=
w 0H
-rH-‘+I
+gVh
0 where (u, v) = v is the velocity vector, H = h + 0, h is the water depth, and < is the surface elevation, V = (a/&c, d/ay), g is the constant gravitational acceleration, f, the Coriolis parameter, Y, the coefficient of bottom friction, and R, a domain with contour dR. On the solid part of the contour, a&, we have % Im, = 0
(2)
where v,, is the velocity normal to dR1. In order to formulate the boundary condition on the open part of contour dRz, let us consider the following characteristic matrix on dRz l?=n,;i+n,fi
(3)
where n,, ny are projections of the external normal n to the boundary. It is known (Lax & Phillips’) that the number of boundary conditions is equal to the number of negative eigenvalues of the characteristic matrix coincident with the number of the ingoing characteristics. We have
r-x1=
i
vn - JJ
0
0
vn - X
Hn,
Hny
gn, gn,
(4)
vn - X )
where I is the identity matrix. Hence X1 = v-n = v,,, X2,3 = v,, f c, c = &i7 and in the subsonic case at v,, I c one should set two boundary conditions at the inflow points (i.e. when v and n have opposite signs) and one at the outflow points. The form of these conditions is determined by the requirement of non-negativity of the quadratic form (i, @ I an2 L 0
2.1 Cartesian coordinates
Consider the initial-boundary
shallow-water equations
(5)
where r is a symmetric matrix obtained from r by substitution gH = c*/4, ii = (v, 2~). This inequality ensures an estimate of the solution in some norm,
Open boundary conditions for dynamics of a strait
269
which implies the uniqueness of the solution. The allowable boundary conditions for the quasi-linear hyperbolic system are analyzed in Oliger and SundStrom.” Thus, for system (1) the correct form of the conditions at an open boundary is the following. For X,Y E
dfl2
VT =
for v, < 0
72
w2 = 21,- @C
= 93
(6)
u2 = v, E’ = [, c2 = r).
for 21,> 0
where v, is the tangential velocity. The prescribed functions qi include invariants corresponding to characteristics outcoming from the domain onto the boundary. At the inflow points, invariant - wt = lvn 1+ &$?C and characteristic velolcity v, can be expressed by invariant w2 = ) v, 1- um,
r=t
U u=
21 0
H
U=O
E,rl E a*,
on
CEW
(9)
Since transformation (7) does not change the type of the equation at open boundaries the situation with the boundary conditions is the same as that considered above. We have: cosA(n,5) = 0, cos (n, 7) = f 1, r = f 3, i.e. the matrix n,B loo; is a characteristic one. The eigenvalues of B are: X1 = V, X2,3 = V f c fl, g22 = 11,’+ T$ js the corresponding component of the metric tensor elek, i, k = 1,2. At the inflow points VIZ, < 0, i.e. the characteristic matrix has two negative eigenvalues and it is thus necessary to set two boundary conditions; at the outflow points Vn, > 0 only one boundary condition is to be set. To find the form of conditions at an open boundary, let us compose the matrix R-‘, so that its rows are the normed left-hand eigenvectors of matrix R. Transformation R-‘&R diagonalizes R
:i
V R-‘&R=A=
0
i 0
:;/3)
0
0
V+c@ 0
0 V-cfl
t20
(8)
1 (10)
PI
with Jacobian J = 8(x: y)/d(&n), 0 # J < cc. In the new variables (7) the set of eqns (1) has the form ut + Au, + Bu, = F,
Let us turn to the boundary condition. Transformation (7) is assumed to be such that the solid part of the contour as2T, which is the mapping of dR1, constitutes the sides ,$= const. of the rectangle a* and the open part of Xl; which is the mapping of dR2, has the sides n = const. On the lines c = const. the velocity of a fluid particle is perpendicular to the normal to the boundary: v - 05 = 0, i.e.
R-t=[i
Let us now introduce the curvilinear coordinates c, n concordant with the configuration of Cl: on a chosen segment of Xl one of the coordinates is fixed, while the other is distributed arbitrarily but monotonically. In the I, n-plane, the domain R is now represented by a rectangle a*. Let us consider the transformations E = 5(X,Y)>
where U i = ve’ are the contravariant components, e’ = V{’ is the contravariant basis; i = 1,2; U ’ = U,
s = qx/fl,
I = q,,/@.
Multiplying eqn (10) by R-’
we write R-+I, + A(R-‘II), = ‘p = R-l@ - Au 0), n,X2 < 0. Thus, in this case: 21,= yi, V + c p g h -1 C = 72. Other situations may be examined similarly. For computations it is more suitable to use the equations obtained by transition to the contravariant flow components: p = JHU, q = JHV. Multiplying the equation of motion by ei = VO
(24)
and analogous smoothing for 6. On introducing eqn (24) into eqn (20) eqn (21) will contain viscosity immedi-
271
+ JU(I+w)
= F-e2
(25)
where w = J-‘(QF - I’,), whence, with regard to Q = g2i U’, the q* is obtained. The structure of matrix A allows
