# Computation of Implicit Representation of Volumetric Shells with Predefined Thickness
## 3 Second-Order Offset Function  
A first-order accurate solution to the offset problem is only acceptable if \(h\) is very small. In what follows, we present a systematic approach to rise the order of accuracy starting from \(\phi^{(1)}(\vec{x},h)=0\).  
Let us first note that since \(\phi^{(1)}(\vec{x},h)\) is an implicit surface itself, we are able to apply the same methodology again in order to obtain an offset function of \(\phi^{(1)}(\vec{x},h)\). In the following recursive computation of the offset function \(\phi_{n}^{(m)}(\vec{x},h)\), we will indicate the level of the recursion by subscript \(n\) and the order of approximation by superscript \((m)\). With this, it is, for example, possible to cover the distance \(h\) in two steps of equal size. To achieve this, we recursively apply Equation (3) in itself to obtain \[\phi_{2}^{(1)}(\vec{x},h) = \phi_{1}^{(1)}(\vec{x},h/2)-\frac{h}{2}\sqrt{(\
abla\phi_{1}^{(1)}( \vec{x},h/2))^{2}}. \tag{4}\]  
This second iteration \(\phi_{2}^{(1)}(\vec{x},h)\) has the same order of accuracy as \(\phi^{(1)}(\vec{x},h)\), as can be seen by substituting \(\phi_{1}^{(1)}(\vec{x},h/2)=\phi(\vec{x},h/h)+\mathcal{O}(h^{2})\) and applying the triangle inequality while realizing that \(h\) is not a function of \(\vec{x}\), i.e., \(\sqrt{(\
abla(\phi(\vec{x},h/h)+\mathcal{O}(h^{2})))^{2}}\leq\sqrt{(\
abla( \phi(\vec{x},h/h))^{2}}+\mathcal{O}(h^{2})\).  
While \(\phi_{2}^{(1)}(\vec{x},h)\) has the same order of accuracy as \(\phi^{(1)}(\vec{x},h)\), it is expected to be more accurate as it uses half the step size. Adding more steps would raise the accuracy further but the methodology would quickly become unpractical as the algebraic complexity of the implicit surface function grows with each step. Instead, it appears more promising to combine the obtained solutions through Richardson extrapolation ([PERSON], 1966). Richardson extrapolation is a convergence acceleration method for sequences of approximations obtained with different smallness parameters (e.g., step sizes). It is applicable if the order of convergence with respect to the smallness parameter is either known or can be obtained directly from the series of solutions. For this, it must be possible to express the error of the method as a Taylor series in the smallness parameter. It is also necessary that the error does not depend on terms not related to the smallness parameter. Therefore, only results obtained by the same deterministic method can be combined in a Richardson extrapolation.  
We recall the general equation for Richardson extrapolation given a solution \(g^{(q)}(h)\) obtained with smallness parameter \(h\) and a solution \(g^{(q)}(\beta h)\) obtained with a smallness parameter \(\beta h\), both with the same method of order \(q\):  
\[g^{(q+1)}=\frac{g^{(q)}(h)-\beta^{-q}g^{(q)}(\beta h)}{1-\beta^{-q}}. \tag{5}\]  
In the current case, the refinement factor is \(\beta=1/2\) and the convergence order is \(q=1\). Note that both methods have to be evaluated at the same final location, which in the current case, is after a distance \(h\). However, the solution \(g^{(q)}(h)=\phi_{1}^{(1)}(\vec{x},h)\) is obtained using a single step while the second solution \(g^{(q)}(h/2)=\phi_{2}^{(1)}(\vec{x},h)\) uses two steps of size \(h/2\). After some algebra, this yields  
\[\phi^{(2)}(\vec{x},h)=2\phi_{2}^{(1)}(\vec{x},h)-\phi_{1}^{(1)}(\vec{x},h)=f (\vec{x})-h\sqrt{(\
abla\phi_{1}^{(1)}(\vec{x},h/2))^{2}}. \tag{6}\]  
A second-order approximation of the surface \(\Gamma_{h}^{(2)}\) is obtained from \(\phi^{(2)}(\vec{x},h)=0\).