1. Introduction
We consider numerical solving the hyperbolic equation
equipped with suitable initial and boundary conditions for known velocity coefficient 
Among the successful numerical methods for solving this equation we mention such nonoscillatory conservative finite difference shemes as TVD (total variation diminishing), TVB (total variation bounded), ENO (essentially nonoscillatory), and CABARET (Compact Accurate Boundary Adjusting high REsolution Technique) shemes (see [1]- [11] and the reference there). This approach uses the traditional difference approximation of temporal derivative and is liable to Courant — Friedrichs — Lewy (CFL) condition for ratio between steps in time and space.
The other approach focuses on the approximation of the whole operator of problem (1) as the partial derivative in some direction of the space 
This approach is developed under different names: methods of trajectories or modified characteristics and semi-Lagrangian one (see, for example, [12]- [19] and the reference there).
In order to highlight the essential ingredients of suggested approach we shall operate with one-dimensional problem again, keeping in mind that we shall extend this method in subsequent papers. In this paper, we will first show how to consider the possible boundary conditions in contrast to the periodic conditions of the previous paper [16]. Then we modify the presented algorithm for big and huge velocity and give a numerical example that demonstrates the stability independent of CFL condition.
2. The statement of problem and the main theorem
So, in the rectangle 
consider equation
(1)
Assume that coefficient 
is given at 
and is positive for simplicity sake.
Unknown function 
satisfies the initial condition
(2)
and boundary condition
(3)
Assuming the continuity of 
at we get 
at the coordinate origin. Moreover, for continuity of first partial derivatives in (1) the following equality must hold:
By simple calculations we can obtain the necessary condition for the continuity of the second partial derivatives at the point (0, 0), etc. We will not dwell on the question of the sufficiency of this conditions for the smoothness of the solution 
and at once we assume sufficient smoothness of the velocity 
and solution for further considerations.
Let us take two time lines 
with 
and two nodes 
For both these nodes we construct the characteristics 
of equation (1) at segment 
[20, 21]. They satisfy the ordinary differential equation with different initial values:
(4)
Fig. 1. Trajectories in standard (first) situation
These characteristics define two trajectories 
in plane 
when 
The typical situation is considered in the previous paper [20] when each of these trajectories crosses line 
in some point 
We supposed that they are not mutually crossed and therefore 
For this case the following result was proved in [16].
Theorem 1. For smooth solution of equation (1) in the standard (first) situation (Fig. 1) we have equality
(5)
But the boundary condition (3) may produce two other situations. First, for small 
the trajectory 
may be interrupted at line 
at point 
because function is unknown for 
But trajectory 
continues up to line 
(see Fig. 2). And second, both trajectories and are interrupted at line at points and 
(see Fig. 3). We enumerate these situations from one to three.
Fig. 2. Second situation: trajectory C1 is interrupted and C2 does not
Fig. 3. Third situation: both trajectories C1 and C2 are interrupted
For last situations we prove two equalities.
Theorem 2. For smooth solution of equation (1) with boundary condition (3) in the second situation (Fig. 2) we get the equality
(6)
Proof. Define by 
the curvilinear pentagon bounded by lines 
And define by 
the corresponding parts of these lines, which form the boundary 
(see Fig. 4). Introduce also the external normal 
defined at each part of boundary except 5 vertices of pentagon.
Fig. 4. Integration along the boundary in second case
Now use formula by Gauss-Ostrogradskii [20, 21] in the following form:
(7)
where sing 
means scalar product. Since the boundary 
consists of five parts we calculate the integral over separately on each line:
(8)
Along the line 
the external normal equals 
Then
(9)
At arbitrary point 
the tangent vector is 
Therefore the external normal (that is orthogonal to it) equals
(10)
It implies equality
(11)
Along the line 
the external normal equals 
Then
(12)
For other two parts of the boundary we use the same way to calculate the integrals:
(13)
(14)
Combination of (7) — (9) and (11) — (14) implies (6):
?
The next result is justificated like previous theorem with some simplification. Therefore we give it without any proof.
Theorem 3. For smooth solution of equation (1) with boundary condition (3) in the third situation (Fig. 3) we get the equality
(15)
Note that this situation contains the case when 
and coincides with 
So, consideration of the boundary conditions in the calculation expressions 
resulted in an additional calculation of the integrals along 
of known function 
In this sense, consideration of boundary conditions makes minor modifications into the algorithms discussed below, so in further considerations we return to the periodic case.
3. Piece-wise linear discrete approximation
So, let the condition of periodicity holds for the problem (1) — (2), namely functions 
are supposed periodical in 
with period 
and are smooth enough for further considerations.
Now we formulate some modification of numerical algorithm from [16] for solving problem (1) — (2). First, take integer 
and construct uniform mesh in 
with nodes 
and meshsize 
Than take integer 
and construct uniform mesh in with nodes 
and meshsize 
Then make the following cycle for 
supposing that the approximate solution 
is known yet at previous time-level for 
1. With the help of values in these points and periodicity we construct the piecewise linear (periodical) interpolant
(16)
2. For each point 
construct approximate trajectories 
down to time-level 
for example, by Runge-Kutta method. They produce cross-points 
If 
goes outside segment 
we use periodicity of our data. One can see that we get values which do not coincide with exact values 
Let we solve equations (4) with the following accuracy:
(17)
where 
is small enough.
3. For each interval 
compute integral
(18)
by trapezoid quadrature formula separately at each nonempty subinterval 
where 
is linear.
4. Due to Theorem 1 it is supposed that
(19)
Finally we put
(20)
Thus, we complete our cycle which may be repeated up to last time-level 
Condensed form of this algorithm in terms of piecewise linear periodical interpolants is written as follows:
(21)
So, we get approximate discrete solution 
at each time-level 
First we prove the conservation law in discrete form.
Let a discrete function 
is given, and we construct piecewise linear interpolant 
with period 1.
Theorem 4. For any initial condition the approximate solution (16) — (20) 
satisfies the equality:
(22)
Proof. The justification directly coincides with the proof of Theorem 2 in [16] with substitution instead of
It is interesting that this discrete conservation law is exactly valid for an approximate values 
Now we prove a stability of algorithm (17) — (20) in the discrete norm analogous to that of space 
(23)
Theorem 5. For any intermediate discrete function 
the solution 
of (17) — (20) satisfies the inequality:
(24)
Proof. Again the justification directly coincides with the proof of Theorem 3 in [16] with substitution instead of
Now evaluate an error of approximate solution in introduced discrete norm.
Theorem 4. For sufficiently smooth solution of problem (1) — (2) we have the following estimate for the constructed approximate solution:
(25)
with a constant 
independent of 
Proof. We prove this inequality by induction in 
For 
this inequality is valid because of exact initial condition (2): 
Suppose that estimate (25) is valid for some 
and prove it for
So, at time-level 
we have decomposition
(26)
with a discrete function 
that satisfies the estimate
(27)
Because of Taylor series in of 
in the vicinity of point 
we get equality
(28)
Because of Theorem 1
Instead of 
let use its piecewise linear periodical interpolant 
Then
(29)
Thus, we get equality
(30)
For 
we use (21) and (26):
(31)
where values of 
are constructed by piecewise linear periodical interpolation.
Now let us evaluate the difference caused by error (17):
From the properties of piecewise linear interpolant it follows that
Therefore
(32)
Now let subtract (31) from (30), multiply its modulus by 
use (32), and sum for all 
(33)
Due to Theorem 3 last term in brackets is combined into 
Thus
(34)
Let put 
then this inequality is transformed with the help (27):
that is equivalent to (27).
We can see that at last time level we get inequality
(35)
In some sense we got a restriction on temporal meshsize 
to get convergence. For example, to get first order of convergence, it is enough to take
with any constant 
independent of 
But this restriction is not such strong for constant as CFL condition:
(37)
Moreover, it is opposite in meaning: here the greater 
the better accuracy.
Thus, this approach is convenient for the problems with huge velocity which come from a computational aerodynamics: we have computational stability on the base of Theorem 4 and conservation law on the base of Theorem 3.
Now discuss the choice of 
in (17). From (35) the better choice is 
i.e., 
For this purpose the standard Runge-Kutta method is acceptable that gives 
with appropriate accuracy and stability conditions [22, 23].
4. Numerical experiment
Let take
and solve the equation (1) with initial condition
subject to periodicity. Then exact solution is
The result of implementing the presented algorithm is given in Table 1. Here in this experiment we set 
(as the most implemented ratio in computations). The first line shows a number n of mesh nodes and the middle line contains the values 
at last time level 
The last line demonstrates the order of accuracy.
Table 1
|
n |
40 |
80 |
160 |
320 |
640 |
1280 |
|
Error |
0.01228 |
0.00712 |
0.00194 |
0.00064 |
0.00026 |
0.00013 |
|
Order of accuracy |
1.341834 |
1.876304 |
1.192391 |
1.971816 |
1.017456 |
Thus we indeed have at least the first order of accuracy on h when 
/ h is fixed.
5. Conclusion
Thus, we continue presentation of the numerical approach [16] which is more convenient for huge velocity Here we show the treatment with boundary condition instead of periodical one, and then we examine theoretically and by numerical example the effect of the approximate solving the characteristics equations instead of exact process.
Again we have to note that the accuracy is the higher the less time steps done in the algorithm. But for the equations with nonzero right-hand side a small time step 
will be crucial for appropriate approximation.
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