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### 二维浅水方程的隐式非线性一致解法 #### 概述本文介绍了一种针对二维浅水方程（含科里奥利力项）的隐式非线性一致(INC)求解技术。该方法利用了隐式格式的无条件稳定性特性，能够在重力波和对流的Courant-Friedrichs-Lewy (CFL)数大于1的情况下使用较大的时间步长。所谓“非线性一致”是指在迭代过程中所有未知变量在同一时间水平上出现，并被同时求解（即不存在时间上的分裂误差）。该INC方法与传统的半隐式方法进行了对比分析，展示了其在系统动力学时间尺度得到充分解决时能够保持较高精度的特点。 #### INC方法的优势 - **稳定性与效率**：由于该方法采用隐式格式，因此不受稳定性限制，可以在保证解的质量的同时使用较大的时间步长。 - **准确性**：INC方法通过同时求解所有未知量，在一定程度上避免了时间分裂误差，从而提高了解的准确性。 - **适应性强**：对于那些具有较大对流CFL数的问题，INC方法依然能够提供准确的解。 #### 方法比较 - **INC方法与半隐式方法**：通过对比INC方法和传统半隐式方法在求解二维浅水方程中的表现，结果显示INC方法即使在对流CFL数大于1的情况下也能维持高精度。为了进一步解释这一结果，研究者进行了近似修正方程分析(AMEA)，量化了两种方法之间的差异。 #### 实现技术 - **JFNK方法**：本研究所使用的INC解法采用了雅各比自由牛顿-克罗洛夫(JFNK)方法。JFNK方法通过避免显式构造雅各比矩阵来提高计算效率。 - **预处理**：为提高JFNK方法的收敛速度，研究中采用了基于半隐式算法的高度矩阵预处理技术。 - **多网格线性求解器**：半隐式算法中的高度矩阵是通过多网格线性求解器进行求解的，这种方法不仅高效而且具有良好的可扩展性。 #### 结论本文提出的隐式非线性一致(INC)方法为解决二维浅水方程提供了一种新的视角。通过对比分析，该方法在保证计算稳定性和准确性方面显示出了明显的优势。特别是对于那些具有复杂动力学特性的物理问题，INC方法通过避免时间上的分裂误差，显著提升了求解的精度。未来的研究可以进一步探索该方法在更高维度问题、更复杂的边界条件以及并行计算环境下的应用潜力。

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OVEMBER

2002 2611MOUSSEAU ET AL.

q 2002 American Meteorological Society

An Implicit Nonlinearly Consistent Method for the Two-Dimensional Shallow-Water

Equations with Coriolis Force

V. A . M

OUSSEAU

,D.A.K

NOLL

AND

J. M. R

EISNER

Los Alamos National Laboratory, Los Alamos, New Mexico

(Manuscript received 9 November 2001, in ﬁnal form 19 April 2002)

ABSTRACT

An implicit and nonlinearly consistent (INC) solution technique is presented for the two-dimensional shallow-

water equations. Since the method is implicit, and therefore unconditionally stable, time steps may be used that

result in both gravity wave Courant–Friedrichs–Lewy (CFL) numbers and advection CFL numbers being larger

than one. By nonlinearly consistent it is meant that all of the unknown variables appear at the same time level

in the equations and are solved for simultaneously in an iterative manner (i.e., no splitting errors in time). The

INC method is compared to a more traditional semi-implicit method for stepping over the gravity wave stability

constraint. Results are presented that show that the second-order-in-time INC method can maintain a high level

of accuracy if the dynamical timescale of the system is resolved by the time step. To investigate this difference

in temporal integration accuracy between the nonlinearly consistent method and the semi-implicit method an

approximate modiﬁed equation analysis was performed. The INC solution technique employed is the Jacobian-

free Newton–Krylov (JFNK) method. Preconditioning of the JFNK method is achieved via a semi-implicit

solution method. The height matrix from the semi-implicit algorithm is solved using a multigrid linear solver

to provide efﬁciency and scalability.

1. Introduction

An implicit nonlinearly consistent (INC) solution

method is presented for the solution of the shallow-

water equations including the Coriolis force in two-di-

mensions. We demonstrate that a second-order-in-time

INC method can provide accurate solutions to problems

even when the advective Courant–Friedrichs–Lewy

(CFL) number is greater than one. To explain this result

we performed an approximate modiﬁed equation anal-

ysis (AMEA) to quantify the differences between an

INC solution method and a traditional semi-implicit

method. The results of this AMEA study indicate that

the semi-implicit method is ﬁrst order in time but the

error terms are different than the error terms in the ﬁrst-

order INC method.

In this initial study the semi-implicit method is chosen

for comparison because of its ease of implementation

and the simplicity of its modiﬁed equation analysis.

Both the semi-implicit semi-Lagrange method (Stani-

forth and Cote 1991) and the split explicit method

(Klemp and Wilhelmson 1978), which are more com-

monly used in numerical weather prediction, are obvi-

ous choices for future studies. These future studies could

Corresponding author address: Dr. V. A. Mousseau, Los Alamos

National Laboratory, T-3, Fluid Dynamics, MS B216, Los Alamos,

NM 87545.

E-mail: vmss@lanl.gov

include the same components used in the current com-

parison, a modiﬁed equation analysis and time step con-

vergence study, and a simultaneous comparison of ac-

curacy and efﬁciency (how much CPU time is required

for a given level of accuracy).

The approach that we employ to achieve a INC meth-

od is the Jacobian-free Newton–Krylov (JFNK; Brown

and Saad 1990; Chan and Jackson 1984) technique. In

this approach, Newton’s method provides the nonlinear

consistency and a Krylov method is used to solve the

large implicit matrix. To improve the efﬁciency of the

JFNK method we employ a ‘‘physics-based’’ precon-

ditioner. In this physics-based approach, the precondi-

tioner is constructed from a time split solution method

(a semi-implicit method in this study). Within our semi-

implicit preconditioner we use a multigrid linear solver

(Briggs 1987). As a result, the amount of work, per

control volume, per time step, is constant in the solution

algorithm.

This work is a continuation of earlier work on the

application of JFNK methods to geophysical ﬂows

(Reisner et al. 2001). In this earlier work the one-di-

mensional shallow-water equations and the two-dimen-

sional Navier–Stokes equations were solved but the

JFNK method was not used on the entire nonlinear sys-

tem. In this manuscript we solve the two-dimensional

shallow-water equations with Coriolis force. This equa-

tion set was chosen for two reasons. First, this equation

set can be used to represent important phenomenon in

2612 V

OLUME

130MONTHLY WEATHER REVIEW

both atmospheric (Enagonio and Montgomery 2001)

and oceanic (Nadiga et al. 1997; Milliff and McWilliams

1994) applications. Second, it provides a simple equa-

tion system that contains multiple timescales. The fastest

timescale in the shallow water equations (SWE) is as-

sociated with the gravity wave. The second timescale

of interest results from ﬂuid advection. This is the typ-

ical timescale that is tracked by numerical methods to

ensure stability and accuracy. A third timescale exists

in our system that is the bulk motion of a vortex. The

timescale associated with the translation of the vortex

will be referred to as the dynamical timescale of the

problem and it will always be the slowest timescale in

this study. The vortex translational velocity will be set

by including a source term in the momentum equations

that represents a background wind. The fourth timescale

is associated with the Coriolis frequency and the im-

portance of resolving this timescale will be shown in

section 6.

The semi-implicit method is a popular solution meth-

od for solving the SWEs because of its stability, speed,

and accuracy (Thomas and Browning 2001). The basic

idea behind the semi-implicit method is to implicitly

solve for the effects of the gravity waves and explicitly

track the advection timescale. This method is popular

because the implicit part of the algorithm removes the

time step stability restriction based on the gravity wave

speed and replaces it with a stability limit based on the

ﬂuid velocity (Casulli 1990; Skamarock et al. 1997).

From a stability point of view, the semi-implicit method

allows one to take much larger time steps than an ex-

plicit method.

The semi-implicit method only requires the solution

of a matrix with one unknown per control volume

(height) to advance in time a system with three un-

knowns per control volume (height, x velocity, y ve-

locity). This scalar matrix, the pressure or height matrix,

is the only implicit contribution to the solution method.

The height matrix is constructed by substituting the two

momentum equations into the continuity equation to

obtain a single equation for the ﬂuid height. Because of

multigrid’s linear scaling, a semi-implicit method using

multigrid as the linear solver provides a fast solution

algorithm (Barros et al. 1990; Bates et al. 1990; Ruge

et al. 1995, 2000; Adams and Smolarkiewicz 2001).

However, the shallow-water equations are a nonlinear

system of equations. The main sources of nonlinearity

are from the momentum ﬂux terms and the gradient of

potential energy term in the conservative form of the

momentum equation. Consider two distinct approaches

to solving a set of nonlinear equations. The ﬁrst ap-

proach is to linearize the problem and then take time

steps small enough to keep the linearization error small.

This is effectively what is done in the semi-implicit

method. The second approach is to iterate within the

time step until the nonlinear error is small. This will be

referred to as a nonlinearly consistent solution method.

This study will compare the accuracy and efﬁcacy (ac-

curacy as a function of work) of these two approaches

for solving the shallow-water equations.

We will brieﬂy discuss the three main components

that have allowed Newton’s method to become a viable

option in both the solution of nonlinear systems and in

4D data assimilation (Li et al. 1994): 1) matrix-free

implementations, 2) an inexact Newton’s method, and

3) Krylov solutions of the linear problems.

A more recent version of Newton’s method, JFNK

(Brown and Saad 1990; Chan and Jackson 1984), does

not require the formation and storage of the Jacobian

matrix, which was always one of the major obstacles to

using Newton’s method. The action of the Jacobian ma-

trix can be approximated with a ﬁnite difference (Brown

and Hindmarsh 1986), similar to approximating the ac-

tion of the Hessian matrix [see Navon and Legler

(1987), Eq. (133)], used in data assimilation. Since the

Jacobian (or Hessian) matrix is not formed, this is re-

ferred to as matrix free. In both data assimilation and

nonlinear solution methods it is known that accurate

solutions of the linear problem are not required until the

nonlinear iteration is near convergence. This approach

of modifying the linear convergence tolerance is re-

ferred to as an inexact Newton method (Dembo et al.

1982) in the nonlinear solver community but it is similar

to the truncated Newton method (Nash 1984) used to

solve the large-scale unconstrained minimization prob-

lems of 4D data assimilation. Krylov solvers have been

employed by the 4D data assimilation community (Na-

von and Legler 1987) and the nonlinear solver com-

munity as an efﬁcient method to solve the large matrices

that result from both applications. In this study we use

the generalized minimal residual (GMRES; Saad and

Schultz 1986; Saad 1996) algorithm as the Krylov meth-

od. The GMRES method is a conjugate-gradient-like

solver that is applicable to nonsymmetric matrices.

The Jacobian-free Newton–Krylov part of the ma-

chinery is well developed for both the nonlinear solution

of systems of equations and for data assimilation. How-

ever, the key to an efﬁcient JFNK method for solving

systems of nonlinear equations is preconditioning. We

are developing a concept referred to as ‘‘physics-based’’

preconditioning (Mousseau et al. 2000; Knoll and

Mousseau 2000). The general motivation of physics-

based preconditioning is that any solution method that

provides a smooth and reasonable approximation to a

time step for the problem can be employed as a pre-

conditioner. The solution method employed as a pre-

conditioner in this manuscript is a semi-implicit tech-

nique. The height matrix required by the semi-implicit

method has only a single unknown per control volume

(compared to three unknowns per control volume in the

Jacobian matrix [h, u, and

]). Within our preconditioner

this matrix is approximately inverted using a simpliﬁed

multigrid method (Mousseau et al. 2000).

For clarity, we restate the new contributions contained

in the manuscript. First, an approximate modiﬁed equa-

tion analysis is used to demonstrate that time splitting

OVEMBER

2002 2613MOUSSEAU ET AL.

solution methods, like the classic semi-implicit method,

have ﬁrst-order temporal error terms that are different

from an INC solution procedure. Second, with a second-

order INC solution method, one can take time steps on

the order of the dynamical timescale (which may be

larger than the advective stability limit) and still achieve

accurate solutions to the shallow-water equations.Third,

inside of the JFNK method one can use a semi-implicit

solution method as a preconditioner, resulting in an

overall algorithm that is fast, accurate, and scalable for

the solution of the shallow-water equations.

The remainder of the paper has the following orga-

nization. Section 2 presents an approximate modiﬁed

equation analysis for both a semi-implicit method and

an INC method for time integration of the two-dimen-

sional shallow-water equations with Coriolis force in

conservative form. Following this, in section 3, the INC

solution method of this paper, physics-based precon-

ditioned JFNK, is presented in detail. Section 4 presents

the discrete form of the nonlinear residuals and section

5 presents the details of the physics-based precondi-

tioner. Then in section 6 results are presented that dem-

onstrate the accuracy of the solution method, the scal-

ability of the method, and that the JFNK method is more

efﬁcient than the semi-implicit method for a given level

of accuracy in test problems where there is a wide spread

between the fastest and slowest timescales. Finally in

section 7 conclusions and future work are discussed.

2. An approximate modiﬁed equation analysis of

time integration of two methods

To compare two time integration techniques we em-

ploy a modiﬁed equation analysis (MEA; Hirt 1968;

Warming and Hyett 1974). Instead of doing the MEA

on the discrete equations we will perform the MEA on

the semi-discrete equations where the spatial derivatives

are still in differential form. This modiﬁcation to the

MEA will be referred to as an (AMEA). In this section

an AMEA is performed that illuminates the difference

between an implicit nonlinearly consistent solution

method and the classic semi-implicit method. For related

work on using AMEA to analyze time splitting errors

in different algorithms, see Knoll et al. (2002, manu-

script submitted to J. Comput. Phys.). The semi-implicit

method (which splits the time step and uses variables

at different time levels) has additional ﬁrst-order trun-

cation terms as compared to an implicit nonlinearly con-

sistent method. We are able to show through results in

section 6 that the differences between the semi-implicit

and the nonlinearly consistent method are errors in the

semi-implicit method.

The shallow-water equations with Coriolis force in

conservative form in two dimensions are

]h ]uh ]

1150, (1)

]t ]x ]y

]uh ]uh ]u

h 1 ]h

11 1g 2 f (

)h 5 0, (2)

]t ]x ]y 2 ]x

]

h ]u

h ]

h 1 ]h

111g 1 f (u 2 u )h 5 0. (3)

]t ]x ]y 2 ]y

Here h is the height of the ﬂuid, u is the velocity in the

x direction,

is the velocity in the y direction, g is the

acceleration due to gravity, f is the Coriolis parameter

(frequency), and u

and

are the x and y components

of the background wind.

In this section we will compare the differences in the

temporal integration between the semi-implicit method

and a ﬁrst-order INC method. It will become clear later

in this section that comparing the temporal differences

between two ﬁrst-order methods is an effective approach

since they both have the same leading error terms. Com-

paring the accuracy of these two methods will be left

to section 6. Since the reason for this AMEA is to com-

pare two temporal discretization approaches and not to

address spatial contributions to the error, the spatial dis-

cretizations will not be included.

a. First-order implicit nonlinearly consistent

The semidiscrete versions of equations (1)–(3) for the

ﬁrst-order implicit nonlinearly consistent solution are

n11 nn11 n11

h 2 h ](uh ) ](

h )

1150, (4)

Dt ]x ]y

n11 nn11 n11 n11 n11

uh 2 uh ](uuh ](

Dt ]x ]y

n112

g ](h )

n11 n11

12f (

h 2

h ) 5 0, (5)

2 ]x

n11 nn11 n11 n11 n11

h 2

h ](

h ) ](u

h )

Dt ]y ]x

n112

g ](h )

n11 n11

11f (uh 2 uh ) 5 0. (6)

2 ]y

The approximate modiﬁed equation analysis will use

the following Taylor series expansions:

n1122n11

]h Dt ] h

nn11 3

h 5 h 2Dt 11O(Dt ), (7)

]t 2 ]t

n1122n11

]uh Dt ] uh

nn11 3

uh 5 uh 2Dt 11O(Dt ), (8)

]t 2 ]t

n1122n11

]

h Dt ]

nn11 3

vh 5

h 2Dt 11O(Dt ). (9)

]t 2 ]t

When these Taylor series expansions are substituted into

the semidiscrete versions of the equations and the orig-

inal partial differential equations are gathered on the

left-hand side one gets the following results:

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