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na.numerical evaluation – Smoothly connecting PDEs with finite variations Answer

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na.numerical evaluation – Smoothly connecting PDEs with finite variations

A PDE with non-smooth inhomogeneity

Let $mathcal{L}$ breathe a second-order, linear, elliptic differential operator appearing on $mathcal{C}^2([0,2]^2)$.

I’m numerically fixing the inhomogeneous PDE
commence{align*}
mathcal{L}u(x,y)+(x-1)^+=0,
aim{align*}

the place $(cdot)^+$ denotes the optimistic sever.

Put in a different way, I decipher two PDEs which necessity to breathe related at $x=1$:
commence{align*}
commence{instances}
mathcal{L}u(x,y)=0 & textual content{for }(x,y)in[0,1]occasions[0,2],
mathcal{L}u(x,y)+x-1=0& textual content{for }(x,y)in(1,2]occasions[0,2].
aim{instances}
aim{align*}

Approximating all partial derivatives by central variations, I get the nine-point stencil
commence{align*}
c_1 u_{i-1,j-1} + c_2 u_{i,j-1} + c_3 u_{i+1,j-1} + c_4 u_{i-1,j} + c_5 u_{i,j} &
+ c_6 u_{i+1,j}+ c_7 u_{i-1,j+1} + c_8 u_{i,j+1} + c_9 u_{i+1,j+1} + (x_i-1)^+ &=0.
aim{align*}

Thus, $u$ is the answer to a system of linear equations.

Problem

Plotting the answer $u$, all of it seems exquisite and consummate. However, a plot of $frac{partial u}{partial x}$ as a duty of $x$ reveals that the spinoff just isn’t {smooth} at $x=1$. The above FD strategy works exquisite with value-matching (the answer $u$ is completely steady) however struggles with smooth-pasting at $x=1$ (the spinoff just isn’t {smooth}).

enter image description here

Question

How do I guarantee smooth-pasting with a finite dissimilarity strategy at $x=1$?


Some of my failed makes an attempt comprise

  • Impose that route and backward variations at $x=1$ equal one another (= 2nd bid central dissimilarity is zero).
  • Use increased bid approximations round $x=1$ akin to $u_{xx}approx frac{-u(-2h)+16u(-h)-30u(0)+16u(h)-u(2h)}{12h^2}$ and $u_{x}approx frac{u(-2h)-8u(-h)+8u(h)-u(2h)}{12h}$ and central variations in all places else.
  • Approximating partial derivatives utilizing factors solely from one aspect of $x=1$ (i.e. solely utilizing both $u(0), u(h), u(2h)$ or as a substitute $u(0),u(-h),u(-2h)$).
  • Imposing that $u_xapproxfrac{u_{i+1,j}-u_{i-1,j}}{2h}$ equals an medium of partial derivatives at $1+h$ and $1-h$.

Note: This drawback arises as sever of a bigger system of free border issues. Thus, it is needful to decipher the PDE numerically. This query is too posted right here.

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