**2. NUMERICAL SCHEME**

**2.1 Numerical scheme
for linearized equation**

For the
first step to describe the numerical scheme for the tsunami model, the linearized long wave equation without bottom frictions in
one dimensional propagation, Eq.(2.1),
is introduced.

(2.1)

Let us
introduce the finite difference method to solve the above equation numerically.
The finite difference method based upon the

(2.2)

where *Δt* is the grid interval. We
can form the "forward " difference by
rearranging Eq.(2.2)

(2.3)

where the first term in the right
side of Eq.(2.3) is obviously the finite difference
representation for the first order of time derivative at t*=t*
(see Fig.2.1).

**Figure 2.1**
Central finite difference representations

The truncation error which have the order of *Δt*, (O(*Δt*))
is the difference between the partial derivative and its finite difference
representation. Moreover we can rearrange the Taylor expansion series in Eq.(2.2) by replacing *Δt* by +*Δt* /2 and -*Δt*/2 and then we obtain "central"
difference with the second order of truncation error.

(2.4)

It is
interesting that although the expression of the finite difference
representations in Eqs. (2.3) and (2.4) are similar,
the order of truncation errors are different. By using the above
"central" difference method with the staggered numerical points for
water level and discharges, which is called the staggered leap-frog scheme, we
can descretize Eq.(2.1) as follows.

(2.5)

For dealing with discrete values in numerical computations, *η*(*x,t*) and
*M*(*x,t*) are expressed for the case
of the staggered leap-frog scheme as

(2.6)

where *Δx* and *Δt* are the grid sizes in *x* direction and in time *t*. The point schematics for the
numerical scheme are illustrated in Fig.2.2. The points for
water depth *h* is the same as
those for water elevation, *η*.

**Figure 2.2** The
point schematics for the numerical scheme

The above finite method provide
stable result as long as the C.F.L condition is satisfied:

*C* (celerity) < *Δx* /*Δt*

Details of
the stable condition will be discussed in the Chapter 3.1. Imamura & Goto (1988) investigated the truncation errors in three
kinds of typical scheme for long waves simulations and
showed that in term of numerical accuracy the staggered leap-frog scheme is the
best among them.

**2.2 Numerical scheme
for convection terms**

In the
present numerical scheme, an "upwind" difference scheme is applied to
the convection terms in order to make the computation stable. The reason why
this scheme ensures the stability of computation is explained by taking a
simple convection equation in the following:

(2.7)

Here the coefficient *C*
is the propagation velocity and is assumed constant. The arrangement of
computation points in the present scheme requires the forward difference scheme
for the first order time derivations. This yields

(2.8)

In
addition, the central difference is applied to the space derivative.

(2.9)

As a result, is given by

(2.10)

The
solution of Eq.(2.10) is
implicitly equivalent the solution of Eq.(2.11) with
an truncation error of (*Δt*^{2}+*Δx*^{2}). Substituting Eqs.(2.8) and (2.9) into Eq.(2.7)
yields

(2.11)

If the second-order derivative with respect to time is
rewritten by using the following relationship (this assumption is valid for the
progressive waves),

The
solution of Eq.(2.11) is the
same as the solution of the following diffusion equation in which the diffusion
coefficient is negative.

(2.12)

A negative
diffusion works to amplify round-off errors with time leading to an instability.
Therefore, Eq.(2.10) is an
unstable difference scheme. The more detail about stable and unstable scheme
will be discussed in chapter 3.1.

In order to obtain a stable scheme, the space
derivative term is approximated by either forward or backward difference
depending on the sign of coefficient *C*.
With the forward difference, we have

and with the backward difference

The
corresponding differential equations we are going to solve are within the
truncation error of O(*Δt*^{2}+ *Δx*^{2}),
for the forward difference

(2.13)

and for the backward difference

(2.14)

Therefore,
to keep the virtual diffusion coefficient positive (or say to ensure the
stability of the computation), we have used the backward difference in case of
positive *C*, and the forward
difference in case of negative *C*, in
addition to setting .
In other words, the difference should be taken in the direction of the
flow. This is the reason why this scheme
is called the "upwind" difference.
Although the leap-frog scheme has the truncation error of the order of *Δx*^{2}, as long as the
convection term concerns, its order become large as *Δx*.

**2.3 Numerical scheme
for bottom friction term**

The
friction term becomes a source of instability if it is discretized
with an explicit scheme. To make the discussion of instability simple, let us
consider the following momentum equation without convection terms:

(2.15)

The explicit form of Eq.(2.15) is :

(2.16)

When a
velocity become large or a total depth is small in a very shallow water, the
absolute of coefficient (amplification factor) of the first term on the right
hand side of Eq.(2.16) become more than unity, which
leads to numerical instability. In order to overcome this problem, an implicit
scheme to set a friction term can be basically introduced. For example, a
simple implicit form,

(2.17)

ensures numerical stability,
because the amplification factor in Eq.(2.17) is
always less than unity. However the effect of friction in shallow water becomes
so large that numerical results are dumped. Another implicit form, a combined
implicit one to the friction term is given by,

(2.18)

This scheme also gives a stable result. It is, however, noted
that the above scheme causes a numerical oscillation at the wave front because
the amplification factor could be negative.

We should
select the best scheme among some implicit ones to apply the bottom friction
term with Manning's roughness. Considering the fact that the numerical scheme
of convection terms also involve artificial or numerical dissipation, selection
of Eq.(2.17) causes much
damping in the result. Therefore the present model uses the combined implicit
scheme, Eq.(2.18).