CFD Discretization: Building a Higher Order Code Using Polynomial Fitting

[pcjl]

Hi guys I was wondering if anyone on here could help me out.

Essentially I am trying to build a higher order cfd code and struggling to work out where to start, I’m trying to build it using polynomial fitting so here’s my problem.

Taking my domain as;

ϕ_(i-2)------------------ϕ_(i-1)------------------ϕ_i------------------ϕ_(i+1)
|------------------------|----------------------|---------------------|
x=0---------------------1/3 --------------------2/3------------------x=1

So I’m wanting to fit a 3rd order polynomial however I’ve not really done much polynomial fitting since A level so I’m quite stuck, here’s what I have,

ϕ=ax^3+bx^2+cx+d

Taking the infomation from the domain I can get the four equations

ϕ_(i+1)=a+b+c+d

ϕ_i=a(2/3)^3+b(2/3)^2+c(2/3)+d

ϕ_(i-1)=a(1/3)^3+b(1/3)^2+c(1/3)+d

ϕ_(i-2)=d

My problem is that there seems to be two many unknowns can anyone help

 

[AlephZero]

You have 4 equations in 4 unknowns, so what's the problem solving them?

But you don't actually need to solve them. Look at http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html