## 18.2.1 Solution of the PDF Transport Equation

The PDF has dimensions and the solution of its transport equation by conventional finite-difference or finite-volume schemes is not tractable. Instead, a Monte Carlo method is used, which is ideal for high-dimensional equations since the computational cost increases just linearly with the number of dimensions. The disadvantage is that statistical errors are introduced, and these must be carefully controlled.

To solve the modeled PDF transport equation, an analogy is made with a stochastic differential equation (SDE) which has identical solutions. The Monte Carlo algorithm involves notional particles which move randomly through physical space due to particle convection, and also through composition space due to molecular mixing and reaction. The particles have mass and, on average, the sum of the particle masses in a cell equals the cell mass (cell density times cell volume). Since practical grids have large changes in cell volumes, the particle masses are adjusted so that the number of particles in a cell is controlled to be approximately constant and uniform.

The processes of convection, diffusion, and reaction are treated in fractional steps as described below. For information on the fractional step method, refer to [ 47].

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