The High Order Augmented Finite Volume Methods Based on Series Expansion for Nonlinear Degenerate Parabolic Equations

Abstract

Two high order multi-augmented and improved augmented finite volume methods are proposed for solving nonlinear degenerate parabolic problems. The solution is represented as Puiseux series expansion in a subdomain with singularity, but contains undetermined parameters called augmented variables. The equation in regular subdomain is treated with high accuracy numerical methods and the unknown parameters can be solved simultaneously from the corresponding nonlinear system. The outstanding advantages of the proposed methods are that the degeneracy can be depicted by the semi-analytic solution, and we can get high order results globally. Specially, the convergence order for nonlinear degenerate parabolic problems is determined by the numerical schemes on regular subdomain, and the augmented methods have good robustness for solving degenerate or singular problems. Numerical examples for some degenerate parabolic equations confirm the efficiency of the new methods including the second and fourth order schemes. In particular, a two-dimensional singular elliptic equation with corner degeneracy is presented in the numerical experiments to demonstrate that the proposed methods can be extended to higher dimensional degenerate problems.

Appendices

The derivation of the Equivalent Fredholm Integral Equation in Section 2.1:

Equation (1.1) is equivalent to the following equation

$$\begin{aligned} x^\alpha u_x)_x=u_t-f(x,t,u), (x,t)\in (0,b)\times (0,T], 0\le \alpha <1. \end{aligned}$$

(A.1)

Integrating (A.1) over [x, b], we obtain

$$\begin{aligned} b^\alpha u_x(b,t)-x^\alpha u_x(x,t)=\int _x^b \left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] dx, \end{aligned}$$

(A.2)

which can also be written as

$$\begin{aligned} u_x(x,t)=x^{-\alpha }b^\alpha u_x(b,t)-x^{-\alpha }\int _x^b \left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] dx. \end{aligned}$$

(A.3)

Integrating (A.3) over [0, x], we can obtain that

$$\begin{aligned} u(x,t)-u(0,t)=\frac{x^{1-\alpha }}{1-\alpha }b^\alpha u_x(b,t)-\int _0^x \eta ^{-\alpha }\int _\eta ^b \left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi d\eta . \end{aligned}$$

(A.4)

The integral domain in (A.4) is $D=\{(\eta ,\xi )\mid 0\le \eta \le x, \eta \le \xi \le b\}$. If we change the sequence of integral, the integral domain is $D=\{(\eta ,\xi )\mid 0\le \eta \le \xi , 0\le \xi \le x\} \bigcup \{(\eta ,\xi )\mid 0\le \eta \le x, x\le \xi \le b\}$. So

$$\begin{aligned} \begin{aligned}&\int _0^x \eta ^{-\alpha }\int _\eta ^b \left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi d\eta \\&\quad =\int _0^x \left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi \int _0^\xi \eta ^{-\alpha } d\eta \\&\quad \quad +\int _x^b \left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi \int _0^x\eta ^{-\alpha }d\eta \\&\quad =\int _0^x \frac{\xi ^{1-\alpha }}{1-\alpha }\left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi \\&\quad \quad +\int _x^b\frac{x^{1-\alpha }}{1-\alpha }\left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi . \end{aligned} \end{aligned}$$

(A.5)

Substituting (A.5) into (A.4), and using the boundary condition(1.3), we have

$$\begin{aligned} \begin{aligned}&u(x,t)-g_1(t) \\&\quad =\frac{x^{1-\alpha }}{1-\alpha }b^\alpha u_x(b,t)-\int _0^x \frac{\xi ^{1-\alpha }}{1-\alpha }\left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi \\&\quad \quad -\int _x^b\frac{x^{1-\alpha }}{1-\alpha }\left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi . \end{aligned} \end{aligned}$$

(A.6)

Set $x=b$ in (A.6) yields

$$\begin{aligned} u(b,t)-g_1(t)=\frac{b}{1-\alpha }u_x(b,t)-\int _0^b \frac{\xi ^{1-\alpha }}{1-\alpha }\left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi . \end{aligned}$$

(A.7)

From the boundary condition (1.3), we have

$$\begin{aligned} u(b,t)=g_2(t)-u_x(b,t). \end{aligned}$$

(A.8)

$u_x(b,t)$ can be solved from (A.7) and (A.8) as follows

$$\begin{aligned} u_x(b,t)=\frac{1-\alpha }{1-\alpha +b}[g_2(t)-g_1(t)]+\frac{1-\alpha }{1-\alpha +b}\int _0^b\frac{\xi ^{1-\alpha }}{1-\alpha } \left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi . \end{aligned}$$

(A.9)

Substituting (A.9) into (A.6) yields

$$\begin{aligned} \begin{aligned} u(x,t)=&g_1(t)+\frac{b^\alpha \left[ g_2(t)-g_1(t)\right] }{1-\alpha +b}x^{1-\alpha } \\&+\int _0^x \frac{\xi ^{1-\alpha }}{1-\alpha }\left[ \frac{b^\alpha x^{1-\alpha }}{1-\alpha +b}-1\right] \left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi \\&+\int _x^b \frac{x^{1-\alpha }}{1-\alpha }\left[ \frac{b^\alpha \xi ^{1-\alpha }}{1-\alpha +b}-1\right] \left[ u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))\right] d\xi . \end{aligned} \end{aligned}$$

(A.10)

By introducing the Green’s function

$$\begin{aligned} G(x,\xi )=\left\{ \begin{aligned}&\frac{\xi ^{1-\alpha }}{1-\alpha }\bigg [\frac{b^\alpha }{1-\alpha +b}x^{1-\alpha }-1\bigg ],\ 0\le \xi \le x, \\&\frac{x^{1-\alpha }}{1-\alpha }\bigg [\frac{b^\alpha }{1-\alpha +b}\xi ^{1-\alpha }-1\bigg ],\ x\le \xi \le b, \end{aligned} \right. \end{aligned}$$

(A.11)

the solution of the degenerate parabolic equation (1.1)-(1.3) can be written as the following Fredholm integral equation:

$$\begin{aligned} u(x,t)=g_1(t)+\frac{b^\alpha [g_2(t)-g_1(t)]}{1-\alpha +b}x^{1-\alpha }+\int _0^bG(x,\xi )[u_t(\xi ,t)-f(\xi ,t,u(\xi ,t))]d\xi . \end{aligned}$$

(A.12)

The direct method to obtain (2.7)

From (2.3) we can obtain the first order and second order partial derivative of u(x, t) with respect to x

$$\begin{aligned}&u_x(x,t)=a_0(t)(1-\alpha )x^{-\alpha }+\sum _{j=1}^\infty a_j(t)\alpha _j x^{\alpha _j-1}, \end{aligned}$$

(B.1)

$$\begin{aligned}&u_{xx}(x,t)=a_0(t)(-\alpha +\alpha ^2)x^{-\alpha -1}+\sum _{j=1}^\infty a_j(t)\alpha _j(\alpha _j-1) x^{\alpha _j-2}. \end{aligned}$$

(B.2)

Equation (1.1) is equivalent to

$$\begin{aligned} x^\alpha u_{xx}+\alpha x^{\alpha -1}u_x=u_t-f(x,t,u). \end{aligned}$$

(B.3)

Inserting (B.1), (B.2) and (2.4) into (B.3) yields

$$\begin{aligned} \sum _{j=1}^\infty a_j(t)\alpha _j(\alpha +\alpha _j-1)x^{\alpha +\alpha _j-2}=\sum _{j=1}^\infty c_j(t)x^{\beta _j}. \end{aligned}$$

(B.4)

Equating the powers and coefficients of the corresponding terms on both sides of (B.4) yields

$$\begin{aligned} \left\{ \begin{aligned}&\alpha +\alpha _j-2=\beta _j, \\&a_j(t)\alpha _j(\alpha +\alpha _j-1)=c_j(t),\ j=1,2,\cdots . \end{aligned} \right. \end{aligned}$$

(B.5)

namely

$$\begin{aligned} \left\{ \begin{aligned}&\alpha _j=2-\alpha +\beta _j, \\&a_j(t)=\frac{c_j(t)}{(\beta _j+1)\alpha _j},\ j=1,2,\cdots . \end{aligned} \right. \end{aligned}$$

(B.6)

So (B.6) is the second and third equation in (2.7).

  As to the first equation in (2.7), we apply the boundary condition (1.3) at $x=b$ to (2.3) and (B.1), respectively

$$\begin{aligned} \begin{aligned}&u(b,t)=g_1(t)+a_0(t)b^{1-\alpha }+\sum _{j=1}^\infty a_j(t) b^{\alpha _j}, \\&u_x(b,t)=(1-\alpha )a_0(t)b^{-\alpha }+\sum _{j=1}^\infty \alpha _j b^{\alpha _j-1}, \\&u(b,t)+u_x(b,t)=g_2(t). \end{aligned} \end{aligned}$$

(B.7)

And we can obtain the first equation of (2.7) solved from (B.7)

$$\begin{aligned} a_0(t)=\sum _{j=1}^\infty \frac{c_j(t)b^{\beta _j+1}(\alpha -b-\beta _j-2)}{(1-\alpha +b)(\beta _j+1)(2-\alpha +\beta _j)} +\frac{b^\alpha (g_2(t)-g_1(t))}{1-\alpha +b}. \end{aligned}$$

(B.8)