Mathematical Physics - Volume II - Numerical Methods

Chapter 4. Finite volume method

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literature [24]. Dissipative terms are usually represented by a combination of the second and fourth order terms, ie. D ( W ) i , j , k = ( D 2 x + D 2 y + D 2 z − D 4 x − D 4 y − D 4 z ) W i , j , k . (4.9) In the expression (4.9) the term D 2 x W i , j , k is determined by the relation D 2 x W i , j , k = d ( 2 ) i + 1 / 2 , j , k − d ( 2 ) i − 1 / 2 , j , k , (4.9.1) where is d ( 2 ) i + 1 / 2 , j , k = ε ( 2 ) i + 1 / 2 , j , k h i + 1 / 2 , j , k ∆ t ∆ x W i , j , k , (4.9.2) while the term D 4 x W i , j , k is calculated on the basis of the expression D 4 x W i , j , k = d ( 4 ) i + 1 / 2 , j , k − d ( 4 ) i − 1 / 2 , j , k , (4.9.3) in which is d ( 4 ) i + 1 / 2 , j , k = ε ( 4 ) i + 1 / 2 , j , k h i + 1 / 2 , j , k ∆ t ∆ 3 x W i , j , k . (4.9.4) In the relations (4.9.2) and (4.9.4) terms ∆ x W i , j , k and ∆ 3 x W i , j , k represent the difference operators “forward” oriented ∆ x W i , j , k = W i + 1 , j , k − W i , j , k (4.9.5) and ∆ 3 x W i , j , k = W i + 2 , j , k − 3 W i + 1 , j , k + 3 W i , j , k − W i − 1 , j , k . (4.9.6) The values of the remaining dissipative terms will be determined in a similar way in the expres sion (4.9). Quantities ε ( 2 ) i + 1 / 2 , j , k and ε ( 4 ) i + 1 / 2 , j , k , present in relations (4.9.2) and (4.9.4), are calculated as follows: ε ( 2 ) i + 1 / 2 , j , k = k ( 2 ) max ( ν i + 1 , j , k , ν i , j , k ) , (4.9.7) ie. ε ( 4 ) i + 1 / 2 , j , k = max ( 0 , k ( 4 ) − ε ( 2 ) i + 1 / 2 , j , k ) , (4.9.8) where the quantity ν i , j , k is defined by the expression ν i , j , k = | (4.9.9) In the relations (4.9.7) and (4.9.8) constants k ( 2 ) = O ( 1 ) and k ( 4 ) determine the amount of artificial viscosity added and allow the transition from a fourth-order difference scheme to the scheme of the second order accuracy in the zone of abrupt change of pressure, ie. in the immediate vicinity of the shock wave, which can be deduced from expression (4.9.8). Their size also determines stability and accuracy of numerical schemes [46]. Higher value of constants k ( 2 ) and k ( 4 ) increases the stability of the numerical scheme making the solution smooth, reducing at the same time its accuracy in the zone of abrupt changes of flow variables. p i + 1 , j , k − 2 p i , j , k + p i − 1 , j , k | | p i + 1 , j , k + 2 p i , j , k + p i − 1 , j , k | .