PISO algorithm
PISO algorithm (Pressure-Implicit with Splitting of Operators) was proposed by Issa in 1986 without iterations and with large time steps and a lesser computing effort. It is an extension of the SIMPLE algorithm used in computational fluid dynamics to solve the Navier-Stokes equations. PISO is a pressure-velocity calculation procedure for the Navier-Stokes equations developed originally for non-iterative computation of unsteady compressible flow, but it has been adapted successfully to steady-state problems. PISO involves one predictor step and two corrector steps and is designed to satisfy mass conservation using predictor-corrector steps. Algorithm  steps. The algorithm can be summed up as follows: As already seen for the SIMPLE algorithm, the steps 4 and 5 can be repeated for a prescribed number of times to correct for non-orthogonality. Predictor step Guess the pressure field formula_1 and get velocity field components formula_2 and formula_3 using discretized momentum equation. The initial guess for the pressure may or may not be correct. Corrector step 1Velocity component obtained from predictor step may not satisfy the continuity equation, so we define correction factors p',v',u' for the pressure field and velocity field. Solve the momentum equation by inserting correct  pressure field formula_4 and get the corresponding correct velocity components formula_5 and formula_6. formula_7 formula_8 formula_9 where ;formula_10 :correct pressure field and velocity component formula_11 :correction in pressure field and correction in velocity components formula_12 :guessed pressure field and velocity component We define formula_11 as above. By putting the correct pressure field formula_4 into the discretized momentum equation we get the correct velocity components formula_6 and formula_5. Once the pressure correction formula_17 is known we can find the correction components for the velocity: formula_18 and formula_19. Corrector step 2 In piso another corrector step can be used. formula_20 ; formula_21 formula_22  ; formula_23 formula_24 ; formula_25 where : formula_26 are the correct pressure field and the correct velocity components, respectively and formula_27 are second corrections to the pressure and velocity field. Set formula_28 where; formula_29 are correct pressure and velocity field