A method for numerical solution of a multidimensional convection-diffusion problem
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A METHOD FOR NUMERICAL SOLUTION OF A MULTIDIMENSIONAL CONVECTION–DIFFUSION PROBLEM V. A. Prusov,a A. E. Doroshenko,b and R. I. Chernyshc
UDC 004.75
We propose a modification of the additive splitting algorithm to solve the convection–diffusion problem using an efficient finite-difference scheme. The modification decreases the number of data exchanges and their amount during the numerical solution of a system of multidimensional equations. Approximation, stability, and convergence are considered. Keywords: additive splitting algorithm, convection–diffusion problem, convergence. INTRODUCTION To solve applied problems involving models of environmental processes, researchers should have a grasp of not only the physics of the medium and process but also mathematics. The processes in the Earth atmosphere are usually modeled by nonlinear three-dimensional second-order partial differential equations [1–4]. To obtain practically significant results, numerical methods are applied depending on the complexity of mathematical models. It is often a challenge to choose an optimal solution algorithm because obtaining more realistic results involves greater computational resources and time and needs an improved model and additional conditions. The limiting criteria are usually computational resources and time. Hence, it is necessary to find a trade-off between the accuracy of the solution and the cost. A compromise may be the application of splitting methods to solve multidimensional problems [5–9]. Indeed, to be solved directly, a multidimensional problem needs a great amount of resources and time because of operations on high-dimensional sparse matrices. Splitting the original problem into several subproblems resolves this difficulty but introduces an error into the solution [10]. Based on the specific features of atmospheric models, we may state that even low-accuracy numerical methods can be used because the errors introduced into the model at the stage of formulation and development are usually greater than the numerical errors. Therefore, applying high-accuracy numerical methods without improving the model will not produce noticeably better results. We propose a modification of the additive-averaged componentwise splitting algorithm [11]. Since it allows parallel computations, the modification reduces the number of data exchanges and their amount during the solution. Such a splitting preserves the main advantages of the numerical method [8, 12, 13] (“running” calculation and satisfactory stability conditions), whose combination makes the solution algorithm for a multidimensional problem quite efficient. DEVELOPING A SPLITTING SCHEME Consider a p-dimensional initial–boundary-value convection–diffusion problem p ¶ u p (k ) ¶ u ¶ æ (k ) ¶ u çm + åu =å ¶ t k =1 ¶ x k k =1 ¶ x k çè ¶ xk
u( 0, x1 ,K , x p ) = u 0 for ( x1 ,K , x p ) ÎW ,
ö ÷÷ + f , ø
u( t , x1 ,K , x p ) = u G for ( x1 ,K , x p ) Î G,
(1)
a
National Taras Shevchenko University of Kyiv, Kyiv, Ukraine, [email protected]. bNational Technical University of Ukr
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