Молодой учёный

“Young Scientist” . #3 (50) . March 2013 Mathematics
МАТЕМАТИКА
Characteristics-like approach for solving hyperbolic equation of first order
Синь Вэнь, студент
Университет Бейхан, г. Пекин (Beihang University)
Вяткин Александр Владимирович, кандидат физико-математических наук, научный сотрудник
Институт вычислительного моделирования Сибирского отделения РАН (г. Красноярск)
Шайдуров Владимир Викторович, доктор физико-математических наук, профессор
Институт вычислительного моделирования Сибирского отделения РАН (г. Красноярск), Университет Бейхан, г. Пекин (Beihang University)
1. Introduction
We consider numerical solving the hyperbolic equation
∂r ∂(
ur)
+ = 0, ( tx , ) ∈D,
∂t ∂x
equipped with suitable initial condition for known velocity coefficient u(t, x).
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), and ENO (essentially nonoscillatory)
ones (see, for example, [1]–[14] and the reference there).
In order to highlight the essential ingredients of suggested approach we begin with one-dimensional problem, keeping in
mind that we shall extend these methods in subsequent papers. Moreover, for simplification we take periodic data to avoid a
description complication for inessential issues connected with boundary-value conditions.
2. The statement of problem and the main theorem
Thus, in the rectangle ( tx , ) ∈ D= [0, T]
× [0, 1) consider equation
∂r ∂(
ur)
+ = 0, ( tx , ) ∈D,
∂t ∂x
with initial condition
r(0, x) = r ( x), x∈
[0,1).
(2)
init
Coefficient xtu ),( is given at D and functions r init and u are supposed periodical in x with period 1 and are smooth
enough for further considerations.
One of difficulties in solving (1)–(2) is that solution may contain discontinuities even for smooth data. But we start our
considerations for the case of smooth solution.
Let us take two time lines tk, tk-1[0, T]
∈ with tk - tk-1= t > 0 and two nodes ( tk, x1), ( tk, x2) with points x1, x2∈[0 th points x1, x2∈ [0,1], x1 < x2
(see Fig.1).
For both these nodes we construct the characteristics x (t), i =1, 2, i of equation (1) at segment t∈ [ tk-1, tk]
[16, 17].
They satisfy the ordinary differential equation with different initial values:
⎧dx
i
⎪ = utx ( , i),
t∈[ tk-1, tk],
⎨ dt
⎪
⎩xi(
tk) = xi.
5
(1)
(3)