beq.song {boussinesq}

Song et al.'s analytic solution to Boussinesq equation in a 1D semi-infinite domain with a Dirichlet boundary condition

Description

Song et al.'s analytic solution to Boussinesq equation in a 1D semi-infinite domain with a Dirichlet boundary condition

Usage

beq.song(t = 0.5, x = 1, s = 0.4, h1 = 1, ks = 0.01, nmax = 4, alpha = 1)

Arguments

t

time coordinate.

x

spatial coordinate. Default is seq(from=0,to=L,by=by).

s

drainable pororosity (assumed to be constant)

h1

water surface level or boundary condition coefficient at x=0. Left Dirichlet Bounday Condition.

ks

Hydraulic conductivity

nmax

order of power series considered for the analytic solution solution. Default is 4.

alpha

\alpha exponent see Song at al, 2007

Value

The water surface eletion vs time and space obtained by the analytic solution of Boussinesq Equation

Note

For major details, see Song at al, 2007

Author(s)

Emanuele Cordano

References

Song, Zhi-yao;Li, Ling;David, Lockington. (2007), "Note on Barenblatt power series solution to Boussinesq equation",Applied Mathematics and Mechanics, https://link.springer.com/article/10.1007/s10483-007-0612-x ,doi:10.1007/s10483-007-0612-x

See Also

beq.song.dimensionless

Examples

L <- 1000
x <- seq(from=0,to=L,by=L/100)
t <- c(4,5,20) #  days 

h_sol1 <- beq.song(t=t[1]*3600*24,x=x,s=0.4,h1=1,ks=0.01,nmax=10,alpha=0)
h_sol2 <- beq.song(t=t[2]*3600*24,x=x,s=0.4,h1=1,ks=0.01,nmax=10,alpha=0)
h_sol3 <- beq.song(t=t[3]*3600*24,x=x,s=0.4,h1=1,ks=0.01,nmax=10,alpha=0)
	
	
plot(x,h_sol1,type="l",lty=1,
main="Water Surface Elevetion (Song at's solution) ",
xlab="x[m]",ylab="h[m]")
lines(x,h_sol2,lty=2)
lines(x,h_sol3,lty=3)
legend("topright",lty=1:3,legend=paste("t=",t,"days",sep=" "))