Молодой учёный

“Young Scientist” . #3 (50) . March 2013 Mathematics
max ut≤ h.
(37)
[0,1]
Moreover, it is opposite in meaning: here the greater t the better accuracy.
Thus, this approach is convenient for the problems with huge velocity u(t, x) which come from a computational aerodynamics:
we have computational stability on the base of Theorem 3 and conservation law on the base of Theorem 2.
4. Numerical experiment
Let take u(t, x) =100t, T =1.0 and solve this equation with initial condition
r π
2
init ( x) = sin (2 x), x∈[0,1).
Then exact solution is
r r
2
( tx , ) = init ( x-50 t), x∈[0,1).
The result of implementing the presented algorithm is given in Table 1 for several t and h The first column of Table 1 expresses
a relation t/ h (for most implemented values in computations) and the first string shows a number n of mesh nodes.
Other entries contain the value
t/ h
n
Table 1
64 128 256 512 1024 2048
4 0,02358 0,01203 0,00608 0,00305 0,00153 0,00077
2 0,04536 0,02360 0,01204 0,00608 0,00305 0,00153
1 0,08423 0,04546 0,02362 0,01204 0,00608 0,00305
1 / 2 0,14610 0,08442 0,04548 0,02362 0,01204 0,00608
1 / 4 0,22493 0,14643 0,08446 0,04549 0,02362 0,01204
Thus we indeed have the first order of accuracy on h when t/ h is fixed.
5. Conclusion
Thus, we present the numerical approach which is more convenient for huge velocity u(t, x) then approaches listed in introduction.
Of course, we stay some open questions like boundary condition instead of periodical one, nonlinear dependence
of velocity u on solution r and other. Of course, we need to trace the effect of the approximate solving the characteristics equations
instead of exact process. But we successively consider these issues in next publications, including generalization for twodimensional
and three-dimensional equations.
At first glance, this approach is some integral version of the characteristics method. Moreover, its accuracy is higher, the
less time steps done in the algorithm. But in the future, we will apply it to the equations with nonzero right-hand side for the
approximation of which a small time step t will be crucial.
References:
1. Harten A.: On a class of high resolution total-variation-stable finite-difference schemes // SIAM J. Numer. Anal. –
1984. – V. 21. – P. 1–23.
2. Harten A., Osher S.: Uniformly high-order accurate non-oscillatory schemes // I. SIAM J. Numer. Anal. – 1987. –
V. 24. – P. 279–309.
3. Harten A., Engquist B., Osher S., Chakravarthy S.: Uniformly high-order accurate non-oscillatory schemes, III // J.
Comput. Phys. – 1987. – V. 71. – P. 231–303.
4. Osher S.: Convergence of generalized MUSCL schemes // SIAM J. Numer. Anal. – 1985. – V. 22. – P. 947–961.
5. Osher S., Chakravarthy S.: High-resolution schemes and entropy condition // SIAM J. Numer. Anal. – 1984. – V.
21. – P. 955–984.
6. Osher S., Tadmor E.: On the convergence of different approximations to scalar conservation laws // Math. Comput. –
1988. – V. 50. – P. 19–51.
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