Kernel interpolation - YuriOku/1D_SPH GitHub Wiki

In this section, we will discuss two methods of calculating the spatial derivative of the kernel function $\nabla_i W$ .

Standard gradient

One method is to compute the derivative

$\nabla_i W\left(r_{ij}, h \right) = \frac{\partial r_{ij}}{\partial \mathbf{r}i} \frac{\partial W(r{ij}, h)}{\partial r_{ij}} = \mathbf{e}{ij} \frac{\partial W(r{ij}, h)}{\partial r_{ij}}$

,where $r_{ij} = \left| \mathbf{r}_i - \mathbf{r}j \right|,\ \mathbf{e}{ij} = \frac{\mathbf{r}_i - \mathbf{r}_j}{\left|\mathbf{r}_i - \mathbf{r}_j \right|}$ . For $\frac{\partial W}{\partial r}$ , see Kernel Functions.

Integral Approach

The Integral Approach (IA), proposed by García-Senz et al. (2012), can handle Rayleigh-Taylor instability and Kelvin-Helmholtz instability with better accuracy than the conventional SPH method. First, we have the integral

$I(\mathbf{r}) = \int \left(f(\mathbf{r}^') - f(\mathbf{r}) \right) \left(\mathbf{r}^' - \mathbf{r} \right) W\left(\left|\mathbf{r}^' - \mathbf{r} \right|, h \right) d{r^'}^d \cdot\cdot\cdot\cdot\cdot (1)$

, where $d$ is the dimension of space. Taylor expansion of $f(\mathbf{r}^')$ around a point $\mathbf{r}$ gives

$f(\mathbf{r}^') - f(\mathbf{r}) = \nabla f (\mathbf{r}) \cdot (\mathbf{r}^' - \mathbf{r}) + \mathcal{O}({\mathbf{r}^'}^2) \cdot\cdot\cdot\cdot\cdot (2)$

We ignore the higher-order terms, substitute Eq. (2) into Eq. (1), and consider solving for $\nabla f(\mathbf{r})$ .

1D

In one dimension, equation (1) becomes

$I(r) = \frac{\partial f(r)}{\partial r} \int \left(r^' - r \right)^2 W\left(\left|r^' - r \right|, h \right) dr^' \cdot\cdot\cdot\cdot\cdot (3)$

. Solving equation (3) for $\frac{\partial f(r)}{\partial r}$ and substituting equation (1) for $I(r)$ , we get

$\frac{\partial f(r)}{\partial r} = \frac{\int \left(f(r^') - f(r) \right) \left(r^' - r \right) W\left(\left|r^' - r \right|, h \right) dr^'}{\int \left(r^' - r \right)^2 W\left(\left|r^' - r \right|, h \right) dr^'} \cdot\cdot\cdot\cdot\cdot (4)$

. By using the kernel approximation, we can replace this integral with the kernel sum

$\frac{\partial f(r_i)}{\partial r} = \frac{\sum_j dV_j \left(f(r_j) - f(r_i) \right) \left(r_j - r_i \right) W\left(\left|r_j - r_i \right|, h \right)}{\sum_j dV_j \left(r_j - r_i \right)^2 W\left(\left|r_j - r_i \right|, h \right)} \cdot\cdot\cdot\cdot\cdot (5)$

. This derivative is correct when the field is linear. For example, the derivative of the density field

$\rho_j = \rho_i + a(r_j - r_i)$

$\begin{align*} \frac{\partial \rho_i}{\partial r} &= \frac{\sum_j dV_j \left(\rho_j - \rho_i \right) \left(r_j - r_i \right) W\left(\left|r_j - r_i \right|, h \right)}{\sum_j dV_j \left(r_j - r_i \right)^2 W\left(\left|r_j - r_i \right|, h \right)}\ &= \frac{\sum_j dV_j a \left(r_j - r_i \right)^2 W\left(\left|r_j - r_i \right|, h \right)}{\sum_j dV_j \left(r_j - r_i \right)^2 W\left(\left|r_j - r_i \right|, h \right)}\ &= a \end{align*}$

. This method is equivalent to the linear-exact gradient (Price 2004; Rosswog 2015). However, this method does not conserve momentum because the equation of motion is not antisymmetric for the exchange of $i,,j$ . To be antisymmetric, the equation of motion must be of the form

$\begin{align*} &\frac{\partial f_i}{\partial r} = \sum_j f_j G_{ij}\ &(G_{ij} = -G_{ji}) \end{align*}$

. In the Integral Approach, we assume

$\sum_j dV_j \left(\mathbf{r}_j - \mathbf{r}_i \right) W\left(\left|\mathbf{r}_j - \mathbf{r}_i \right|, h \right) = 0$

. This is the assumption that the distribution of particles is unbiased and uniform. Using this assumption, equation (1) can be expressed as

$I(\mathbf{r}) \sim \sum_j dV_j f(\mathbf{r}_j) \left(\mathbf{r}_j - \mathbf{r}_i \right) W\left(\left|\mathbf{r}_j - \mathbf{r}_i \right|, h \right) \cdot\cdot\cdot\cdot\cdot (7)$

. As a result, the derivative (5) becomes

$\frac{\partial f(r_i)}{\partial r} = \frac{\sum_j dV_j f(r_j) (r_j - r_i ) W\left(\left|r_j - r_i \right|, h \right)}{\sum_j dV_j (r_j - r_i )^2 W\left(\left|r_j - r_i \right|, h \right)}$

. By comparing with the usual derivative in SPH

$\nabla_i f(r_i) = \sum_j f(r_j) dV_j \nabla_i W(| r_i - r_j|, h)$

, the spatial derivative of the kernel function in the Integral Approach

$\nabla_i W\left(\left| r_i - r_j \right|, h\right) = \frac{(r_j - r_i ) W\left(\left|r_j - r_i \right|, h \right)}{\sum_j dV_j (r_j - r_i )^2 W\left(\left|r_j - r_i \right|, h \right)}$

can be obtained.

3-D

Equation (1) in three dimensions is

$\left[ \begin{array}{c} I_1 (\mathbf{r})\ I_2 (\mathbf{r})\ I_3 (\mathbf{r}) \end{array} \right] = \int \left( \left[ \begin{array}{c} \partial f(\mathbf{r})/\partial x_1\ \partial f(\mathbf{r})/\partial x_2\ \partial f(\mathbf{r})/\partial x_3 \end{array} \right] \cdot \left[ \begin{array}{c} x_1^' - x_1\ x_2^' - x_2\ x_3^' - x_3 \end{array} \right] \right) \left[ \begin{array}{c} x_1^' - x_1\ x_2^' - x_2\ x_3^' - x_3 \end{array} \right] W\left(\left| \mathbf{r}^' - \mathbf{r} \right|, h \right) d{r^'}^3$

, where $\mathbf{r} = x_1 \mathbf{i} + x_2 \mathbf{j} + x_3 \mathbf{k}$ and $I(\mathbf{r}) = I_1 (\mathbf{r})\mathbf{i} + I_2 (\mathbf{r})\mathbf{j} + I_3 (\mathbf{r})\mathbf{k}$ . Solving this for $\nabla f(\mathbf{r})$ , we obtain

$\left[ \begin{array}{c} \partial f(\mathbf{r})/\partial x_1\ \partial f(\mathbf{r})/\partial x_2\ \partial f(\mathbf{r})/\partial x_3 \end{array} \right] = \left[ \begin{array}{ccc} \tau_{11} & \tau_{12} & \tau_{13}\ \tau_{21} & \tau_{22} & \tau_{23}\ \tau_{31} & \tau_{32} & \tau_{33} \end{array} \right]^{-1} \left[ \begin{array}{c} I_1 (\mathbf{r})\ I_2 (\mathbf{r})\ I_3 (\mathbf{r}) \end{array} \right]$

, where

$\tau_{ij} = \int (x_i^' - x_i)(x_j^' - x_j) W\left(\left| \mathbf{r}^' - \mathbf{r}\right|, h\right) d{r^'}^3$

and expressing it as a kernel sum, we get

$\tau_{ij} = \sum_b dV_b (x_{i,b} - x_{i,a})(x_{j,b} - x_{j,a}) W\left(\left| \mathbf{r}_b - \mathbf{r}_a\right|, h\right)$

. The inverse matrix in the first term on the right-hand side of equation (12) is

$C = \frac{1}{\tau_{11}\tau_{22}\tau_{33} - \tau_{11}\tau_{23}\tau_{32} + \tau_{12}\tau_{21}\tau_{33} -\tau_{12}\tau_{23}\tau_{31} + \tau_{13}\tau_{21}\tau_{32} - \tau_{13}\tau_{22}\tau_{31}} \left[ \begin{array}{ccc} \tau_{22}\tau_{33} - \tau_{23}\tau_{32} & \tau_{13}\tau_{32} - \tau_{12}\tau_{33} & \tau_{12}\tau_{23} - \tau_{13}\tau_{22}\ \tau_{23}\tau_{31} - \tau_{21}\tau_{33} & \tau_{11}\tau_{33} - \tau_{13}\tau_{31} & \tau_{13}\tau_{21} - \tau_{11}\tau_{23}\ \tau_{21}\tau_{32} - \tau_{22}\tau_{31} & \tau_{12}\tau_{31} - \tau_{11}\tau_{32} & \tau_{11}\tau_{22} - \tau_{12}\tau_{21} \end{array} \right]$

. The i-component of the integral (1)

$I_i(\mathbf{r}) = \int \left(f(\mathbf{r}^') - f(\mathbf{r}) \right) \left(x_i^' - x_i \right) W\left(\left|\mathbf{r}^' - \mathbf{r} \right|, h \right) d{r^'}^3$

can be approximated by eliminating $f(\mathbf{r})$ from the kernel

$I_i(\mathbf{r}a) = \sum_b dV_b f(\mathbf{r}b) \left(x{i,b} - x{i,a} \right) W\left(\left|\mathbf{r}_b - \mathbf{r}_a \right|, h \right)$

The derivative of the function $f(\mathbf{r})$ can be obtained from equation (12) as

$\frac{\partial f(\mathbf{r}a)}{\partial x_i} = \sum{l = 1}^3 C^{il}I_l(\mathbf{r}_a)$

Substituting Eq. (16) into Eq. (17) and comparing it with the derivative of the usual SPH, the derivative of the kernel function becomes

$\frac{\partial W\left(\left|\mathbf{r}b - \mathbf{r}a\right|, h\right)}{\partial x_i} = \sum{l = 1}^3 C^{il} (x{l,b} - x_{l,a}) W\left(\left|\mathbf{r}_b - \mathbf{r}_a\right|, h\right)$

References

García-Senz, D., Cabezón, R. M., and Escartín, J. A., "Improving smoothed particle hydrodynamics with an integral approach to calculating gradients", Astronomy and Astrophysics, vol. 538, 2012. doi:10.1051/0004-6361/201117939.
Price, D. J., "Magnetic fields in Astrophysics", PhDT, 2004.
Rosswog, S., "Boosting the accuracy of SPH techniques: Newtonian and special-relativistic tests", Monthly Notices of the the Royal Astronomical Society, vol. 448, no. 4, pp. 3628-3664, 2015. doi:10.1093/mnras/stv225.