Linearized Block Implicit (LBI) Method applied to Quasi-1D Flow The Linearized Block Implicit Method (LBI) method was developed by Briley & McDonald to solve the 3-D compressible Navier-Stokes equations by time-marching. The LBI method is particularly well suited to handle large systems of coupled PDEs (such as those that occur in reacting flows and plasmas), resulting from many simultaneous elementary processes.

The basic steps of the LBI method are: The LBI method is best illustrated via the governing equations of quasi-1D flow. The basic steps of the LBI method are: Crank-Nicolson difference in time, Central difference in space, Linearize using chain-rule differentiation, and solve the resulting block tridiagonal matrix

Non-dimensionalized governing equations

Evaluate equations at time level (n+1/2), with the intent of relating quantities at (n+1) to their respective values at n. This is done at each point i, but the i is dropped in the difference equations below, for convenience. Continuity or conservation of mass:

Next, use Taylor’s series to evaluate the time derivative: Subtracting, 

Next, evaluate the spatial derivative: Similarly, Adding,

 Thus, the conservation of mass equation becomes: Now,

Using the chain rule for partial differentiation, we have: discretizing the time derivative yields:

Next, apply central differencing for the spatial derivative: and from which we have:

Therefore, Thus, the conservation of mass equation becomes:

Defining , , , , and , the continuity equation becomes:

Next, apply the LBI method to conservation of momentum: In this equation, the first and third terms are handled exactly as in the case of conservation of mass. The second term is handled using the same procedure, but let us look at it in detail:

As before, Using the chain rule for partial differentiation, we have: discretizing the time derivative yields:

Next, apply central differencing for the spatial derivative:

Thus, the conservation of momentum equation becomes:

In the same manner, the energy equation becomes:

Before describing how these equations can be solved, let us modify them to add artificial or numerical dissipation for stability. As before, we will add a term , to the right hand side of the continuity equation, , to the right hand side of the momentum equation, and , to the right hand side of the energy equation.

Each of these terms would have to be evaluated at time level n+1/2. So, and 

 or, and

Thus, similarly, and

The three discretized conservation equations including the artificial dissipation terms, can be written as: where Bi, Di, and Ai are 3x3 matrices for each i, and Fi is a 3x1 column vector. They are given by:

Note that the elements of matrix Bi are the same as the elements of Ai, except that the index (i+1) is replaced by (i-1), and the elements of Bi are negative of the elements of Ai (except for the numerical dissipation terms). The resulting linear system of equations for all i’s is a Nx(N+2) block tri-diagonal matrix, where there are N interior points:

Where. is the vector of unknowns Where is the vector of unknowns. Note that in order to have an NxN matrix, Y0, YN+1, B1, and AN must be eliminated using boundary conditions. This is done by re-defining D1, A1, BN, and DN: and Now, Y0 can be related to Y1 and/or Y2 via boundary conditions:  and

Thus, the first row of the block tri-diagonal matrix is then replaced with and , respectively. In a similar manner YN+1 can also be related to YN and/or YN-1 via boundary conditions:  and General guidance for formulating boundary conditions can be obtained using the method of characteristics, or by physical intuition. Different types of boundary conditions are discussed next.