Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

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having in mind the relation f i = f + i + f − i + 1 . The first term on right hand of the equation (4.120) corresponds to the numerical flux of the first order accuracy, while the remaining terms make a correction of a higher order of accuracy. Solutions of the system of Euler equations can be discontinuous functions, which in mathematical sense satisfy integral form of these equation. However, in addition to discontinuous solutions accompanied by entropy growth in the case of shock waves, there may be such, which are in contradiction with the second principle of thermodynamics. Mentioned solutions, followed by negative entropy increment, must be discarded, ie. such a mechanism that would prevent their emergence and enable obtaining acceptable solutions that satisfy the condition, in literature known as the entropy condition [34], has to be be introduced. Godunov [11] defined the monotonic numerical scheme that prevents oscillations of numerical solution, which is of great importance for obtaining an acceptable solution in the vicinity of the shock wave. In the paper [12] it is shown that the numerical solution of monotonic scheme satisfies the entropy condition. Unfortunately, such a solution is of first-order accuracy, which represents a significant limitation in practical application. A condition weaker than the monotonicity is provided by limitation of the total variation TV of numerical solution, ie. TV ( U ) ≡ ∑ i | U i + 1 − U i | . (4.121) The numerical scheme is said to be total variation diminishing if the relation is satisfied TV ( U n + 1 ) ≤ TV ( U n ) , (4.122) where the indices n + 1 and n refer to the two consecutive moments of integration. The condition (4.122) is sufficient to provide the convergence of the numerical solution in the case of higher order accuracy schemes, but in contrast to the condition of monotonicity does not provide satisfaction of entropy condition . Mentioned condition (4.122) prevents the arising of new local extremes while reducing values of existing local maximum and increasing local minimum of the numerical solution. Fulfillment of conditions (4.122) is provided by introduction of nonlinear limiters , whose basic purpose is to limit abrupt changes of additional terms in numerical flux (4.120). One of the possible choices of nonlinear limiters Ψ leads to the expression for the numerical flux f ( 2 ) i + 1 / 2 = f i + 1 / 2 + m ∑ j = 1 1 − k 4 Ψ + j ( − 1 , 1 ) r ( j ) i − 1 / 2 + 1 + k 4 Ψ + j ( 1 , − 1 ) r ( j ) i + 1 / 2 − − m ∑ j = 1 1 − k 4 Ψ − j ( 3 , 1 ) r ( j ) i + 3 / 2 + 1 + k 4 Ψ − j ( 1 , 3 ) r ( j ) i + 1 / 2 , (4.123) where limiter Ψ ± j ( ℓ, n ) is defined by the expression Ψ ± j ( ℓ, n ) = minmod ( σ ± j , i + ℓ/ 2 , b σ ± j , i + n / 2 ) , (4.124) where is minmod ( x , y ) = sign ( x ) max { 0 , min [ | x | , y sign ( x )] } , (4.125) i + f − i , ie. f i + 1 / 2 = f +