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P. Liu
(3) Alternating direction implicit method
Alternative direction implicit method (ADI method) is one of the finite
difference methods. It is mainly used to solve parabolic or elliptic
partial differential equations, especially for two-dimensional and higher
dimensional heat conduction and diffusion equations. Traditionally, the
Crank–Nicolson method is used to solve the heat conduction equation,
which is time-consuming. The advantage of ADI is that in each iteration step, the equation solved has a simpler structure, so it is easier
to solve. The application of hydrodynamics is usually two-dimensional
and three-dimensional. Due to the requirement of stability, the time
step is limited by the dimension. The higher the dimension, the smaller
the time step required and the larger the calculation workload. In the
mid-1950s, the American scientist J. Douglas Faires et al. proposed the
alternating direction implicit method to speed up the calculation. For
example, in two-dimensional unsteady equations, the first step is to use
the implicit difference for the derivative of X, while the derivative of
Y-direction is the previous value. In the second step, implicit difference
is used for the derivative of Y, and the derivative of X-direction is the
value of the first step. The advantage of this method is that it has good
stability and enough second-order accuracy. The difference equation is
tridiagonal matrix equation, which is easy to solve. SIMPLE algorithm
is a pressure modified semi implicit iterative method. In 1972, S.V.
Patankar, the American scholar, and D.B. Spalding, the British hydrologist, proposed the algorithm. Now it is widely used in computational
fluid dynamics and computational heat transfer in the world. This algorithm soon became the main method to calculate the incompressible flow
field. Later, this algorithm and various subsequent improvements have
been successfully extended to the compressible flow field. It has become
a numerical method that can calculate the flow of any velocity.
(4) Finite basic solution
It is a numerical method for solving potential flow. In the design of lowspeed aircraft in the aviation industry, potential theory is used to calculate
various aerodynamic parameters, that is, to solve the two-dimensional
or three-dimensional Laplace equation. In classical hydrodynamics, it is
very successful to solve the Laplace equation by superposition of basic
solutions. The main point of this method is to replace the influence of
wing and fuselage on the flow field with the distribution of source, sink,
and dipole. Their strength is determined by boundary conditions, and
the results need to be solved by integral equations. It can be solved in
some simple cases, but it is difficult in general cases. The appearance of
P. Liu
(3) Alternating direction implicit method
Alternative direction implicit method (ADI method) is one of the finite
difference methods. It is mainly used to solve parabolic or elliptic
partial differential equations, especially for two-dimensional and higher
dimensional heat conduction and diffusion equations. Traditionally, the
Crank–Nicolson method is used to solve the heat conduction equation,
which is time-consuming. The advantage of ADI is that in each iteration step, the equation solved has a simpler structure, so it is easier
to solve. The application of hydrodynamics is usually two-dimensional
and three-dimensional. Due to the requirement of stability, the time
step is limited by the dimension. The higher the dimension, the smaller
the time step required and the larger the calculation workload. In the
mid-1950s, the American scientist J. Douglas Faires et al. proposed the
alternating direction implicit method to speed up the calculation. For
example, in two-dimensional unsteady equations, the first step is to use
the implicit difference for the derivative of X, while the derivative of
Y-direction is the previous value. In the second step, implicit difference
is used for the derivative of Y, and the derivative of X-direction is the
value of the first step. The advantage of this method is that it has good
stability and enough second-order accuracy. The difference equation is
tridiagonal matrix equation, which is easy to solve. SIMPLE algorithm
is a pressure modified semi implicit iterative method. In 1972, S.V.
Patankar, the American scholar, and D.B. Spalding, the British hydrologist, proposed the algorithm. Now it is widely used in computational
fluid dynamics and computational heat transfer in the world. This algorithm soon became the main method to calculate the incompressible flow
field. Later, this algorithm and various subsequent improvements have
been successfully extended to the compressible flow field. It has become
a numerical method that can calculate the flow of any velocity.
(4) Finite basic solution
It is a numerical method for solving potential flow. In the design of lowspeed aircraft in the aviation industry, potential theory is used to calculate
various aerodynamic parameters, that is, to solve the two-dimensional
or three-dimensional Laplace equation. In classical hydrodynamics, it is
very successful to solve the Laplace equation by superposition of basic
solutions. The main point of this method is to replace the influence of
wing and fuselage on the flow field with the distribution of source, sink,
and dipole. Their strength is determined by boundary conditions, and
the results need to be solved by integral equations. It can be solved in
some simple cases, but it is difficult in general cases. The appearance of