Adaptive Gradient-Augmented Level Set Method with Multiresolution Error Estimation

Abstract

A space–time adaptive scheme is presented for solving advection equations in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian formulation with backward time integration is coupled with a point value multiresolution analysis using Hermite interpolation. Thus locally refined dyadic spatial grids are introduced which are efficiently implemented with dynamic quadtree data structures. For adaptive time integration, an embedded Runge–Kutta method is employed. The precision of the new fully adaptive method is analysed and speed up of CPU time and memory compression with respect to the uniform grid discretization are reported.

Appendix: Two-Dimensional Hermite Interpolation

Suppose that values of a function $u(x_1,x_2)$ and its derivatives $u_{x_1}$, $u_{x_2}$ and $u_{x_1 x_2}$ are given at four vertices of a square of side $h$, as shown in Fig. 18. We will use superscripts $sw$, $se$, $nw$ and $ne$ to refer to these points and the corresponding values.

Fig. 18

Interpolation cell. Markers $\circ$ show the corner points, where the values of the function $u$ and its derivatives $u_{x_1}$, $u_{x_2}$ and $u_{x_1 x_2}$ are known. Markers $\times$ show the 3 points that appear in mid-point interpolation formulae (42–45).

Let us rescale the coordinates $x_1$, $x_2$:

$$\begin{aligned} \tilde{x}_1 = \frac{x_1-x^{\mathrm{sw}}_1}{h}, \quad \quad \tilde{x}_2 = \frac{x_2-x^{\mathrm{sw}}_2}{h}, \end{aligned}$$

(37)

and define basis functions

$$\begin{aligned} f(\tilde{x}) = 2\tilde{x}^3-3\tilde{x}^2+1, \quad \quad g(\tilde{x}) = \tilde{x}^3-2\tilde{x}^2+\tilde{x} \end{aligned}$$

(38)

The value of $u$ at $(x_1,x_2) \in [x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_1+h]\times [x^{\mathrm{sw}}_2,x^{\mathrm{sw}}_2+h]$ can be estimated using the following ${\mathcal {O}}(h^4)$ accurate formula:

$$\begin{aligned} \tilde{u}(x_1,x_2)= & {} ( u^{\mathrm{sw}} f(\tilde{x}_1) f(\tilde{x}_2) + u^{\mathrm{se}} f(1-\tilde{x}_1) f(\tilde{x}_2) \nonumber \\&+ u^{\mathrm{nw}} f(\tilde{x}_1) f(1-\tilde{x}_2) + u^{\mathrm{ne}} f(1-\tilde{x}_1) f(1-\tilde{x}_2) ) \nonumber \\&+h~ ( u_{x_1}^{\mathrm{sw}} g(\tilde{x}_1) f(\tilde{x}_2) - u_{x_1}^{se} g(1-\tilde{x}_1) f(\tilde{x}_2) \nonumber \\&+ u_{x_1}^{\mathrm{nw}} g(\tilde{x}_1) f(1-\tilde{x}_2) - u_{x_1}^{\mathrm{ne}} g(1-\tilde{x}_1) f(1-\tilde{x}_2) ) \nonumber \\&+h~ ( u_{x_2}^{\mathrm{sw}} f(\tilde{x}_1) g(\tilde{x}_2) + u_{x_2}^{\mathrm{se}} f(1-\tilde{x}_1) g(\tilde{x}_2) \nonumber \\&- u_{x_2}^{\mathrm{nw}} f(\tilde{x}_1) g(1-\tilde{x}_2) - u_{x_2}^{\mathrm{ne}} f(1-\tilde{x}_1) g(1-\tilde{x}_2) ) \nonumber \\&+ h^2~ ( u_{x_1 x_2}^{\mathrm{sw}} g(\tilde{x}_1) g(\tilde{x}_2) - u_{x_1 x_2}^{\mathrm{se}} g(1-\tilde{x}_1) g(\tilde{x}_2) \nonumber \\&- u_{x_1 x_2}^{\mathrm{nw}} g(\tilde{x}_1) g(1-\tilde{x}_2) + u_{x_1 x_2}^{\mathrm{ne}} g(1-\tilde{x}_1) g(1-\tilde{x}_2) ). \end{aligned}$$

(39)

It is straightforward to obtain interpolation formulae for the first and second partial derivatives of $u$ by derivating (39).

The values $\tilde{u}(x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2)$, $\tilde{u}(x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_2+\frac{h}{2})$ and $\tilde{u}(x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2+\frac{h}{2})$, as well as unscaled derivatives required for the error estimate, are also obtained from (39). For multiresolution decomposition (16) and reconstruction (17), we define scaled quantities:

$$\begin{aligned} \begin{array}{llll} u_0^{00} = u^{sw}, &{} u_0^{20} = u^{se}, &{} u_0^{02} = u^{nw}, &{} u_0^{22} = u^{ne}, \\ u_1^{00} = \frac{h}{2} u_{x_1}^{sw}, &{} u_1^{20} = \frac{h}{2} u_{x_1}^{se}, &{} u_1^{02} = \frac{h}{2} u_{x_1}^{nw}, &{} u_1^{22} = \frac{h}{2} u_{x_1}^{ne}, \\ u_2^{00} = \frac{h}{2} u_{x_2}^{sw}, &{} u_2^{20} = \frac{h}{2} u_{x_2}^{se}, &{} u_2^{02} = \frac{h}{2} u_{x_2}^{nw}, &{} u_2^{22} = \frac{h}{2} u_{x_2}^{ne}, \\ u_3^{00} = \frac{h^2}{4} u_{x_1 x_2}^{sw}, &{} u_3^{20} = \frac{h^2}{4} u_{x_1 x_2}^{se}, &{} u_3^{02} = \frac{h^2}{4} u_{x_1 x_2}^{nw}, &{} u_3^{22} = \frac{h^2}{4} u_{x_1 x_2}^{ne}, \end{array} \end{aligned}$$

(40)

as well as

$$\begin{aligned} \begin{array}{ll} \tilde{u}_0^{10} = \tilde{u}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2\right) , &{} \tilde{u}_0^{01} = \tilde{u}\left( x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ &{} \tilde{u}_0^{11} = \tilde{u}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ \tilde{u}_1^{10} = \frac{h}{2} \tilde{u}_{x_1}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2\right) , &{} \tilde{u}_1^{01} = \frac{h}{2} \tilde{u}_{x_1}\left( x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ &{} \tilde{u}_1^{11} = \frac{h}{2} \tilde{u}_{x_1}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ \tilde{u}_2^{10} = \frac{h}{2} \tilde{u}_{x_2}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2\right) , &{} \tilde{u}_2^{01} = \frac{h}{2} \tilde{u}_{x_2}\left( x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ &{} \tilde{u}_2^{11} = \frac{h}{2} \tilde{u}_{x_2}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ \tilde{u}_3^{10} = \frac{h^2}{4} \tilde{u}_{x_1 x_2}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2\right) , &{} \tilde{u}_3^{01} = \frac{h^2}{4} \tilde{u}_{x_1 x_2}\left( x^{\mathrm{sw}}_1,x^{\mathrm{sw}}_2+\frac{h}{2}\right) , \\ &{} \tilde{u}_3^{11} = \frac{h^2}{4} \tilde{u}_{x_1 x_2}\left( x^{\mathrm{sw}}_1+\frac{h}{2},x^{\mathrm{sw}}_2+\frac{h}{2}\right) . \end{array} \end{aligned}$$

(41)

Thus we obtain the following formulae:

$$\begin{aligned} \tilde{u}_0^{10}= & {} \frac{1}{2} \left( u^{00}+u^{10}\right) + \frac{1}{4} \left( u_{x1}^{00}-u_{x1}^{10}\right) , \nonumber \\ \tilde{u}_0^{01}= & {} \frac{1}{2} \left( u^{00}+u^{01}\right) + \frac{1}{4} \left( u_{x2}^{00}-u_{x2}^{01}\right) , \nonumber \\ \tilde{u}_0^{11}= & {} \frac{1}{4} \left( u^{00}+u^{10}+u^{01}+u^{11}\right) \nonumber \\&+ \frac{1}{8} \left( \left( u_{x1}^{00}-u_{x1}^{10}+u_{x1}^{01}-u_{x1}^{11}\right) + \left( u_{x2}^{00}+u_{x2}^{10}-u_{x2}^{01}-u_{x2}^{11}\right) \right) \nonumber \\&+ \frac{1}{16} \left( u_{x_1 x_2}^{00}-u_{x_1 x_2}^{10}-u_{x_1 x_2}^{01}+u_{x_1 x_2}^{11}\right) ,\end{aligned}$$

(42)

$$\begin{aligned} \tilde{u}_1^{10}= & {} - \frac{3}{4} \left( u^{00}-u^{10}\right) - \frac{1}{4} \left( u_{x_1}^{00}+u_{x_1}^{10}\right) , \nonumber \\ \tilde{u}_1^{01}= & {} \frac{1}{2} \left( u_{x_1}^{00}+u_{x_1}^{01}\right) + \frac{1}{4} \left( u_{x_1 x_2}^{00}-u_{x_1 x_2}^{01}\right) , \nonumber \\ \tilde{u}_1^{11}= & {} \frac{3}{8} \left( -u^{00}+u^{10}-u^{01}+u^{11}\right) \nonumber \\&- \frac{1}{8} \left( u_{x_1}^{00}+u_{x_1}^{10}+u_{x_1}^{01}+u_{x_1}^{11}\right) - \frac{3}{16} \left( u_{x_2}^{00}-u_{x_2}^{10}-u_{x_2}^{01}+u_{x_2}^{11}\right) \nonumber \\&- \frac{1}{16} \left( u_{x_1 x_2}^{00}+u_{x_1 x_2}^{10}-u_{x_1 x_2}^{01}-u_{x_1 x_2}^{11}\right) \end{aligned}$$

(43)

$$\begin{aligned} \tilde{u}_2^{10}= & {} \frac{1}{2} \left( u_{x_2}^{00}+u_{x_2}^{10}\right) + \frac{1}{4} \left( u_{x_1 x_2}^{00}-u_{x_1 x_2}^{10}\right) , \nonumber \\ \tilde{u}_2^{01}= & {} - \frac{3}{4} \left( u^{00}-u^{01}\right) - \frac{1}{4} \left( u_{x_2}^{00}+u_{x_2}^{01}\right) , \nonumber \\ \tilde{u}_2^{11}= & {} \frac{3}{8} \left( -u^{00}-u^{10}+u^{01}+u^{11}\right) \nonumber \\&- \frac{3}{16} \left( u_{x_1}^{00}-u_{x_1}^{10}-u_{x_1}^{01}+u_{x_1}^{11}\right) - \frac{1}{8} \left( u_{x_2}^{00}+u_{x_2}^{10}+u_{x_2}^{01}+u_{x_2}^{11}\right) \nonumber \\&- \frac{1}{16} \left( u_{x_1 x_2}^{00}-u_{x_1 x_2}^{10}+u_{x_1 x_2}^{01}-u_{x_1 x_2}^{11}\right) \end{aligned}$$

(44)

$$\begin{aligned} \tilde{u}_3^{10}= & {} - \frac{3}{4} \left( u_{x_2}^{00}-u_{x_2}^{10}\right) - \frac{1}{4} \left( u_{x_1 x_2}^{00}+u_{x_1 x_2}^{10}\right) , \nonumber \\ \tilde{u}_3^{01}= & {} - \frac{3}{4} \left( u_{x_1}^{00}-u_{x_1}^{01}\right) - \frac{1}{4} \left( u_{x_1 x_2}^{00}+u_{x_1 x_2}^{01}\right) , \nonumber \\ \tilde{u}_3^{11}= & {} \frac{9}{16} \left( u^{00}-u^{10}-u^{01}+u^{11}\right) \nonumber \\&+ \frac{3}{16} \left( \left( u_{x_1}^{00}+u_{x_1}^{10}-u_{x_1}^{01}-u_{x_1}^{11}\right) + \left( u_{x_2}^{00}-u_{x_2}^{10}+u_{x_2}^{01}-u_{x_2}^{11}\right) \right) \nonumber \\&+ \frac{1}{16} \left( u_{x_1 x_2}^{00}+u_{x_1 x_2}^{10}+u_{x_1 x_2}^{01}+u_{x_1 x_2}^{11}\right) \end{aligned}$$

(45)

In the notations of (16–17), we obtain

$$\begin{aligned} (\tilde{u}_\iota )_{2j_1+1,2j_2}^l = \tilde{u}_\iota ^{10}, \quad (\tilde{u}_\iota )_{2j_1,2j_2+1}^l = \tilde{u}_\iota ^{01}, \quad (\tilde{u}_\iota )_{2j_1+1,2j_2+1}^l = \tilde{u}_\iota ^{11}, \end{aligned}$$

(46)

where $\iota = 0,\ldots ,3$ and indices $j_1$, $j_2$ and $l$ are determined by $\varvec{x}^{sw}$ and $h$, as described in Sects. 2.2 and 2.3. Note that the computational cost of this procedure is much less than interpolation at an arbitrary point because the coefficients that include the basis functions (38) are pre-computed analytically.