Partial Differential Equations  Method of characteristics
Method of characteristics
Method of characteristics
is a method of numerical integration of systems of partial differential equations of hyperbolic type. For the first time for some special cases it
has been considered in D’Alembert’s proceedings. Now the method of characteristics is widely applied in problems of distribution of waves in
hydro air dynamics. So, dynamics of liquid flow in view of distributed parameters on length of pipeline (pressure, flow speed, mass, viscous
friction) is described by the system of quasilinear hyperbolic partial differential equations [4–6]:
(18)
where p, u – average on section pressure and speed of liquid flow;
t – time; х – coordinate on pipe length;
ρ
– liquid density;
– the sound speed in a liquid in view of elasticity of pipeline
walls; Е_{1} – the volumetric module of elasticity of a liquid; Е_{2} – the module of elasticity of a material
of a pipe; d, δ – internal diameter and thickness of a wall of the pipeline;
Φ
(u, x)
– nonlinear function of viscous friction. For the pipeline of round section
where
λ
– coefficient of hydraulic resistance depending on Reynolds’ number
Re = u d / ν
, here
ν
– kinematic viscosity of a liquid.
Integration of the nonlinear equations (18) we’ll execute by numerical method of characteristics according to which the initial partial
differential equations (18) are replaced with the ordinary differential equations
(19)
taking place along direct and return characteristics dx = ± c dt.
Numerical integration of the equations (19) by method of characteristics is carried out as follows. On plane xt (Fig.4) a grid of
direct and return characteristics is under construction, and a step of integration on length Δx is connected with step of integration
on time Δt by linear correlation: Δx = c Δt. After transition in the equations (19) to finite differences in
view of given boundary conditions we solve the received system of equations by method of iterations.
Fig.4. Construction of direct and return characteristics
Let's consider the problem (18) – (19) solution by method of characteristics at the certain boundary conditions. We’ll enter in nodes of grid
of characteristics (Fig.4) the following designations: p ( x_{i}
,
t_{k}
)
= p_{ik} ;
u ( x_{i} , t_{k} ) = u_{ik };
λ
( x_{i} , t_{k} ) =
λ_{
}_{
ik }
. Then in view of boundary conditions u_{ok }= f
(t_{k} ), u_{nk }= g (t_{k} ) after exchanging in equations (19) derivatives by
finitedifference relations, and variables p, u and
λ
– their average values in the neighbor nodes of grid on direct and return
characteristics [accordingly ( x_{i} , t_{k} ), ( x_{i}_{1}
, t_{k}_{1} ) and ( x_{i} , t_{k} ), ( x_{i}_{+1}
, t_{k}_{1} )], we’ll receive the following system of nonlinear algebraic equations:
(20)
where f (t_{k} ) and g (t_{k} ) – functions of time in boundary nodes of pipeline, defining
external influences acting on a liquid flow (for example, pulsation of flow because of the pump kinematics, volumetric or throttle regulation,
etc.);
λ_{
}_{
ik}
– coefficient of hydraulic resistance in node (
x_{i}
, t_{k} ):
(21)
where.
The received system of equations (20) – (21) is solved by an iterative method.
