数理解析研究所講究録
1129 巻 2000 年 129-138

129

NUMERICAL SIMULATION OF ONE-PHASE STEFANPROBLEMS IN ARBITRARY PRECISION1$\mathrm{T}\mathrm{a}\mathrm{r}\mathrm{m}\mathrm{i}_{\mathrm{Z}}\mathrm{i}^{1}$, Hitoshi Imai (今井仁司)2, Toshiki Takeuchi (竹内敏己)2
and Masahiro Kushida (櫛田雅弘)3

1Graduate School of Engineering, The University of Tokushima

2Faculty of Engineering, The University of Tokushima
3Department of General Education, Anan College of Technology

1Introduction

One-phase stefan problem is a boundary value problem involving differential equations on
domains, parts of whose boundaries, the free boundaries are unknown and must be determined as part of the solution. We can find many problems in practical phenomena as stefan
problems. To solve these problems are not easy because geometries of unknown domains are
not simple and they are nonlinear.
On the other hand, we presented a numerical method which realizes arbitrary precision
numerical simulation to PDE systems with smooth solutions. The method consists of two
method, i.e. multiple precision arithmetic and the spectral collocation method. The spectral collocation method is used for the control of truncation errors and multiple precision
arithmetic is used for control of rounding errors. Both errors can be reduced arbitrarily. In
this paper our method is applied to one-phase stefan problems.

2

Our Method and Test Problems

2.1

Arbitrary Precision Numerical Simulation

Errors in numerical simulations originate from truncation errors in the discretization androunding errors. The realization of arbitrary precision simulations needs arbitrary reduction
1This work is partially supported by Grant-in-Aid for Scientific search( . 09440080 and 10354001),$\mathrm{R}e$

Japan Ministry of Education, Science and Culture.

$\mathrm{N}\mathrm{o}\mathrm{S}$130

of both errors. We presented already a simple method for numerical simulation in arbitrary. Here, truncation
precision to one- and two-dimensional free boundary
errors in the discretization and rounding errors can be reduced arbitrarily.
The truncation errors can be reduced by raising the order of the approximation. The
finite difference method , the finite element method and the boundary element method arevery practical, however, they are not practical as higher order discretization formulae. On
the other hand, the spectral collocation method is suitable as a higher order discretization
formula. Therefore, we have adopted the spectral collocation method as a discretization
method. In particular, the spectral collocation method is used here. This is because itis very useful to nonlinear problems. In the spectral collocation method, the order of theapproximation can be controlled by the number of collocation points. For example, inthe spectral collocation method with the Chebyshev-Gauss-Lobatto points, the N-th order.
approximation can be realized only by using $(N+1)$ collocation
is used for reduction of rounding errors. A lot
In addition, multiple precisionof FORTRAN subroutines about multiple precision arithmetic is already known. We used thelibrary of FORTRAN subroutines on the net (http:.
Our method consists of these two methods, i.e. multiple precision arithmetic and thespectral collocation method. In our method, truncation errors and rounding errors arecontrolled easily. This is very important in numerical simulations in applied mathematics.
Of course, both errors can be reduced arbitrarily.
$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{s}[3,6,9]$$\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{s}[1]$$\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{c}[8]$$//\mathrm{w}\mathrm{w}\mathrm{w}.\mathrm{l}\mathrm{m}\mathrm{u}.\mathrm{e}\mathrm{d}\mathrm{u}/\mathrm{a}\mathrm{c}\mathrm{a}\mathrm{d}/\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}/\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{y}$$/\mathrm{d}\mathrm{m}\mathrm{S}\mathrm{m}\mathrm{i}\mathrm{t}\mathrm{h}/\mathrm{F}\mathrm{M}\mathrm{L}\mathrm{I}\mathrm{B}.\mathrm{h}\mathrm{t}\mathrm{m}\mathrm{l})[2]$

2.2

Test Problems

In this paper, we consider the following two test problems. These test problems are typical
one-phase stefan problems.
Test Problem 1. Find

$u$

and

$s$

such that$u_{t}(x, t)=u_{xx}(x, t)+2(t+2)$ ,$0<t$ ,$u(x, 0)=1-x^{2}$ ,$0<x<s(\mathrm{o})$$u(0, t)=(t+1)^{2}$ ,$0\underline{\backslash }t/$$u(s(t), t)=0$ ,$0\leq t$,$\frac{d}{dt}s(t)=-\frac{u_{x}(s(t}{2}\underline{),t)}-t$$s(0)=1$ .

$0<x<s(t)$ ,,,$0<t$ ,,

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Remark 1. The exact solution to Test Problem 1 is given as follows:$u(x, t)=(t+1)^{2}-x^{2}$ ,$s(t)=(t+1)$ ,Test Problem 2. Find

$u$

and

$s$

$0\leq t$,$0\leq t$.

$0\leq x\leq s(\mathrm{t})$,such that$u_{t}(X, t)= \frac{4(t+1)}{\pi^{2}}u_{xx}(_{X,t})+\frac{\pi xe^{-t}\sin\frac{\pi x}{2\sqrt{t+1}}}{4(t+1)\frac{3}{2}}$$0<t$ ,$u(x, 0)= \cos\frac{\pi}{2}x$,$0<X<s(t)$ ,,$0<x<s(\mathrm{o})$$u(0, t)=e^{-t}$ ,$u(s(t), t)=0$ ,,$\frac{d}{dt}s(t)=-\frac{e^{t}}{\pi}u_{x}(_{S}(t), t)$$0\leq t$,$0\leq t$,$0<\mathrm{t}$,,$s(0)=1$ .

Remark 2. The exact solution to Test Problem 2 is given as$u(x, t)=e^{-t} \cos\frac{\pi x}{2\sqrt{t+1}}$,$s(t)=\sqrt{t+1}$ ,2.3$0\leq t$,$0\leq t$.

$\mathrm{f}o$llows:$0\leq x\leq s(t)$,Fixed Domain Method

The spectral collocation method cannot be applied directly to one-phase stefan problems
because of the restriction on the shape of the domain. Then to avoid this difficulty we use
the fixed domain method which was developed in. A one-phase stefan problem istransformed into a fixed boundary problem using mapping functions.
We use the following variable transformation: $(x, t)arrow(\xi, \tau)$ such that$\mathrm{F}\mathrm{D}\mathrm{M}[7]$

$x$

$=$

$x(\xi, \tau)$

$t$

$=$

$\tau$

,

(2.1)
(2.2)

then
$u_{t}$$=$,$- \frac{\partial(u,x)}{\partial(x,t)}=-\frac{\partial(u,x)}{\partial(\xi,\tau)}\frac{\partial(\xi,\tau)}{\partial(x,t)}=-\frac{1}{J}(u_{\tau}X_{\xi}-u\xi X_{\mathcal{T}})$

(2.3)

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$u_{x}$$=$$u_{xx}$$=$$- \frac{1}{J}u_{\xi}$

(2.4)

,

(2.5)

$\frac{u_{\xi\xi}}{J^{2}}-\frac{u_{\xi^{X}\xi\xi}}{J^{3}}$where$J= \frac{\partial(x,t)}{\partial(\xi,\tau)}=x_{\xi}$

(2.6)

.

In Test Problems 1 and 2, mapping function is given as follows;$x( \xi, t)=\frac{s(t)}{2}(\xi+1)$,$0\leq t$,$-1\leq\xi\leq 1$

.

(2.7)

the time axis is divide into
For application of the spectral collocation method inintervals. In each interval the initial and boundary value problem is solved. This procedure
is executed iteratively. For the application of the spectral collocation method to interval
we consider the following variable transform.
$\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}[4,5]$$[t_{S}, t_{e}]$,$t(\tau)$$=$$\frac{\triangle t}{2}\tau+\frac{1}{2}(t_{S}+t_{e})$$\tau(t)$$=$$\frac{2}{\Delta t}(t-\frac{1}{2}(ts+\mathrm{t}_{e}))$$-1\leq\tau\leq 1$,$\triangle \mathrm{t}=t_{e}-t_{s}$

,

(2.8)
(2.9)

then
$\frac{\partial}{\partial t}=\frac{\partial}{\partial\tau}\frac{d\tau}{dt}=\frac{2}{\triangle t}\frac{\partial}{\partial\tau}$(2.10).

By fixed domain method in space and time, Test Problems 1 and 2 are transformed into
the following Test Problems and respectively.
$2’$

$1’$

Test Problem$1’$. Find$u$and$\tilde{s}$such that$\frac{2}{\triangle t}u_{\tau}(\xi, \tau)=\frac{4}{\{\tilde{S}(\mathcal{T})\}2}u_{\xi\xi}(\xi, \mathcal{T})+\frac{(\xi+1)}{\tilde{s}(\tau)}\{-\frac{u_{\xi}(1,\tau)}{\tilde{s}(\tau)}-\frac{\triangle t}{2}\tau$$- \frac{1}{2}(t_{S}+t_{e})\}u_{\xi}(\xi, \mathcal{T})+\triangle t_{\mathcal{T}}+t_{S}+te+4$,$-1<\mathcal{T}\leq 1$$u(\xi, -1)=u_{s}(\xi)$,$u(1, \tau)=0$ ,,$\frac{2}{\triangle t}\tilde{s}’(\tau)=-\frac{u_{\xi}(1,\mathcal{T})}{\tilde{s}(\tau)}-\frac{\triangle t}{2}\tau-\frac{1}{2}(t_{s}+t_{e})$.

$-1<\xi<1$ ,$-1<\xi<1$ ,,$u(-1, \tau)=\{\frac{\triangle t}{2}\tau+\frac{1}{2}(t_{s}+t_{e})+1\}^{2}$$\tilde{s}(-1)=\tilde{s}_{s}$,$-1\leq\tau\leq 1$,$-1\leq\tau\leq 1$,$-1<\mathcal{T}\leq 1$

,

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Here

and if$\tilde{s}(\tau)=s(t(\tau))$then$t_{s}=0$$u_{s}( \xi)=1-\frac{(\xi+1)^{2}}{4}$Test Problem$2’$. Find$u$and$\tilde{s}$,$\tilde{s}_{s}=1$.

such that$\frac{2}{\Delta t}u_{\mathcal{T}}(\xi, \tau)=\frac{16(\frac{\Delta t}{2}\tau+\frac{1}{2}(ts+t_{e})+1)}{\{\tilde{s}(\mathcal{T})\}^{2}\pi^{2}}u_{\xi\xi}(\xi, \tau)$$- \frac{2(\xi+1)e^{\frac{\Delta t}{2}}\tau+\frac{1}{2}(t_{s}+te)}{\{\tilde{s}(t)\}2\pi}u_{\xi()}1,$$\mathcal{T}u\xi(\xi, \tau)$$+ \frac{\pi\tilde{s}(t)(\xi+1)e-(\frac{\Delta t}{2}\tau+\frac{1}{2}(ts+t_{e}\rangle)}{8(\frac{\Delta t}{2}\tau+\frac{1}{2}(t_{s}+te)+1)^{\frac{3}{2}}}\sin\frac{\pi\tilde{s}(t)(\xi+1)}{4\sqrt{\frac{\triangle t}{2}\tau+\frac{1}{2}(\mathrm{t}_{s}+t_{e})+1}}$$-1<\tau\leq 1$,$u(\xi, -1)=u_{s}(\xi)$ ,$-1\leq\tau\leq 1$,$u(1, \tau)=0$ ,$-1\leq\tau\leq 1$,$-1<\tau\leq 1$ ,$\frac{2}{\triangle t}\tilde{s}’(_{\mathcal{T})}=-\frac{2e^{(\frac{\Delta t}{2}\mathcal{T}+}\frac{1}{2}(t_{S}+t_{\epsilon}))}{\tilde{s}(\tau)\pi}u_{\xi}(1, \mathcal{T}),$

Here

.

$\tilde{s}(\tau)=s(t(\mathcal{T}))$and if$t_{s}=0$

then$u_{s}( \xi)=\cos\frac{\pi}{4}(\xi+1)$3$-1<\xi<1$ ,$-1<\xi<1$ ,$u(-1, \tau)=e^{-t}$ ,$\tilde{s}(-1)=\tilde{s}_{\mathit{8}}$,,$\tilde{s}_{s}=1$.

Numerical Results

In this section, our method is applied to test problems.

3.1

Discretization and the Newton’s Method

By applying the spectral collocation method in space and time to Test Problems and ,we obtain two nonlinear systems of equations, Test Problems 1” and 2”, corresponding to
Test Problems 1’ and 2’.
$1’$

$2’$

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Test Problem 1”.
$\frac{2}{\triangle t}\sum_{l=0}^{N_{t}}(D)_{j,,l}\mathcal{T}lu_{i}-\frac{4}{\{s_{j}\}^{2}}\sum_{k=0}^{N_{x}}(D\xi\xi)i,kuk,j+\frac{z_{i}}{s_{j}}\{\frac{1}{s_{j}}\sum^{N_{x}}k=0(D_{\xi})_{0},ku_{k,j}$$+T_{j} \}\sum^{N_{x}}(D_{\xi})k=0i,ku_{k,j}-(2T_{j}+4)=0$,$(i=1,2, \cdots, N_{x}-1, j=0,1, \cdots, N_{t}-1)$,$\frac{\mathit{2}}{\triangle t}\sum_{l=0}^{N_{t}}(D\tau)_{j,ll}s+\frac{1}{s_{j}}\sum_{k=0}^{N_{x}}(D\xi)0,kuk,j+^{\tau_{j}0}=$$(j=0,1, \cdots, N_{t}-1)$$u_{i,N_{t}}-u_{s}(\xi i)=0$,$(i=1,\mathit{2}, \cdots , N_{x}-1)$$u_{N_{x},j}-(T_{j}+1)^{2}=0$ ,$(j=0,1, \cdots, N_{t})$$u_{0,j}=0$ ,$(j=0,1, \cdots, N_{t})$$s_{N_{t}}-\tilde{S}_{s}=0$

where$(D_{\xi})_{j,l},$Test Problem$(D_{\mathcal{T}})_{j,l}$:$(D_{\xi\xi})_{j,l}$:$2^{\prime J}$

Chebyshev collocation first derevative in
Chebyshev collocation first and second derevative in,$\tau$$u(\xi_{i,j}\tau)$$u_{i,j}$$=$$s_{j}$$=$$\tilde{s}(\tau_{j})$$z_{i}$$=$$(\xi_{i}+1)$$T_{j}$$=$$\frac{\triangle t}{2}\tau_{j}+\frac{1}{2}(t_{S}+te)$

,
,

.

.

$\frac{2}{\triangle t}\sum_{l=0}^{N_{t}}(D\mathcal{T})_{j,l}lu_{i},-\frac{16(T_{j}+1)}{\{s_{j}\}^{22}\pi}\sum_{k=0}^{N_{x}}(D\xi\xi)i,kuk,j$$+ \frac{2z_{i}e^{T_{j}}}{\{s_{j}\}^{2}\pi}\sum_{k=0}^{x}(D_{\xi})_{0}N,ku_{k,j}\sum_{k=0}(D_{\xi})i,kuN_{x}k,j$$- \frac{\pi s_{j^{Z_{i}e}}-^{\tau_{j}}}{8(T_{j}+1)\frac{3}{2}}\sin\frac{\pi s_{j}Z_{i}}{4\sqrt{T_{j}+1}}=0$

,

$\xi$

135

$(i=1,2, \cdots, N_{x}-1, j=0,1, \cdots, N_{t}-1)$$\frac{\mathit{2}}{\triangle t}\sum^{N_{t}}(D_{\mathcal{T}})l=0j,l^{S}l+\frac{2e^{T_{j}}}{s_{j}\pi}\sum_{k=0}^{N_{x}}(D_{\xi})0,kuk,j=0$,$(j=0,1, \cdots, N_{t}-1)$$u_{i,N_{t}}-u_{S}(\xi_{i})$

$=$

$0$

,

$(i=1,2, \cdots, N_{x}-1)$$u_{N_{x\dot{\theta}}}-e-T_{j}$

$=$

$0$

,

$(j=0,1, \cdot\cdot*, N_{t})$$u_{0,j}$

$=$

$0$

,

$(j=0,1, \cdots, N_{t})$

$=$

$0$

.

$s_{N_{t}}-\tilde{S}s$$N_{x}$

andare the order of the spectral collocation method in space and time respectively.
Here, Test Problems
andrepresented more general form as a nonlinear system$N_{t}$$1^{\prime/}$$2^{;}/\mathrm{r}\mathrm{a}\mathrm{e}$$F=F_{k}(X)=0$ ,$i=0,1,$$k=1,\mathit{2},$$\cdots,$$\cdots,$$m$$N_{x}$,$j=0,1,$$\cdots,$$N_{t}$$(m=(N_{x}+2)(Nt+1))$

where
$X$

$=$

$(_{S_{0},S}\underline{1,\cdots,SN_{t},u_{0,0,0}u,1,\cdots,u_{N,N}})^{T}xt$$m$$(x_{1}, x_{2}, \cdots, x_{m})T$$=$$F_{k}$

elements

which related to Test Problem

$1”$

.

is defined as follow:$\frac{2}{\Delta t}\sum_{l=}^{N_{t}}0(D)\mathcal{T}j,l^{S}l+\frac{1}{s_{j}}\sum k=0(N_{x}D_{\xi})_{0,k}uk,j+T_{j}$,$(j=0,1, \cdots, N_{t}-1)$$F_{p}=\{$$s_{j}-\tilde{s}_{S}$where$p=j+1$ ,,$(j=N_{t})$$(1 \leq p\leq N_{t}+1)$$\frac{2}{\triangle t}\sum_{l=}^{N_{t}}\mathrm{o}(D\tau)_{j},lui,l-\frac{4}{\{s_{j}\}^{2}}\sum_{k}^{N}=0(\Phi D_{\xi\xi})i,ku_{k}i$$+ \frac{z}{s}\mathrm{L}\{\frac{1}{s_{j}}\sum^{N_{x}}jk=0(D_{\xi})0,ku_{k},j+\tau j\}\sum^{N_{x}}k=0(D_{\xi})_{i,k}uk,j$$-(\mathit{2}\tau_{j}+4)$$F_{q}=\{$,$(i=1,2, \cdots, N_{x}^{\vee}-1, j=0,1, \cdots, N_{t}-1)$$u_{i,j}-u_{s}(\xi_{i})$,$u_{i,j}-(\tau_{j}+1)2$$u_{i,j}$$(i=1,\mathit{2}, \cdots, N_{x}-1, j=N_{t})$,$(i=N_{x}, j=0,1, \cdots, N_{t})$$(_{\backslash }i=0, j=0,1, \cdots, N_{t})$

136

where

$q=(N_{t}+1)(i+1)+(j+1)$ ,Similarly,$F_{k}$$N_{t}+2\leq q\leq(N_{x}+2)(N_{t}+1)$

which related to Test Problem$2^{\prime/_{\mathrm{i}_{\mathrm{S}}}}$.

defined as follow:$\frac{2}{\Delta t}\sum_{l=}^{N_{t}}0(D)_{j,l}\tau ls+\frac{2e^{T_{j}}}{s_{j}\pi}\sum_{k=}^{N_{x}}\mathrm{o}(D_{\xi})_{0},ku_{k,j}$,$(j=0,1, \cdots, N_{t}-1)$$F_{p}=\{$$(j=N_{t})$$s_{j}-\tilde{s}_{s}$where$p=j+1$ ,$(1 \leq p\leq N_{t}+1)$$\frac{2}{\Delta t}\sum_{l0}^{N_{t}}=(D_{\tau})j,lu_{i,l}-\frac{16(T_{j}+1)}{\{s_{j}\}^{2}\pi^{2}}\Sigma_{k}^{N_{x}}=0(D_{\xi}\xi)_{i},ku_{k},j$$+ \frac{2ze^{T_{j}}}{\{s_{j}\}^{2}\pi}\sum_{k=^{0}}N_{x}(D\xi)0,ku_{k},j^{\sum}k=0Nx(D\xi)_{i,k}uk,j$$- \frac{\pi s_{j}z_{i}e-\tau_{j}}{8(T_{j}+1)\S}\sin\frac{\pi s_{j}z_{i}}{4\sqrt{T_{j}+1}}$$F_{q}=\{$$(i=1,\mathit{2}, \cdots, N_{x}-1, j=0,1, \cdots, N_{t}-1)$$u_{i,j}-u_{s}(\xi_{i})$$u_{i,j}-e^{-T_{j}}$,$(i=1,2, \cdots, N_{x}-1, j=N_{t})$,$(i=N_{x}, j=0,1, \cdots , N_{t})$$(i=0, j=0,1, \cdots, N_{t})$$u_{i,j}$

where

,

$q=(N_{t}+1)(i+1)+(j+1)$ ,$N_{t}+2\leq q\leq(N_{x}+2)(N_{t}+1)$ .

Futhermore, we obtain the iteration scheme by the Newton’s method$X^{(n+1)}=X^{(n)}-A^{-1}F(X^{()}n)$$n=0,1,\mathit{2},$ $\cdots$

where$A= \frac{\partial(F)}{\partial(X)}$,$A^{-1}$

inverse of$A$.

We can also write the iteration scheme$AX^{(n+1)}=AX^{(n)}-F(x^{(}n))$$n=0,1,2,$$\cdots$

then we use the Gauss elimination method to solve simultaneous linear equations.

3.2

Numerical Results

pointsFor test problems, the spectral collocation method with theand 60 digit numbers have been used. For evaluation of the numerical results, the maximum
error is used. The maximum error is difined such that$\mathrm{C}\mathrm{h}\mathrm{e}\mathrm{b}\mathrm{y}\mathrm{S}\mathrm{h}\mathrm{e}\mathrm{V}-\mathrm{G}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{S}- \mathrm{L}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{t}_{0}$$\max_{k}|X_{k}^{()}exact-x_{k}(numer)|,$$(k=1,2, \cdots, m)$

137

where
$X_{k}^{(exaCt})$$X_{k}^{(num}er)$$m=(N_{x}+2)(N_{t}+1)$: the computed value of exact solution at collocation point: the computed value of numerical solution at collocation point: total collocation points
$k$

$k$

Table 1. Maximum errors for Test Problem 1.

For Test Problem 1, we use. Numerical calculation is executed only for the first. Here, we use the same order approximation in space and time for simplicity.
interval
The results are superior in precision. This is because the exact solution to Test Problem 1
is polynomial in space and time.
$\triangle t=1$$[0, \triangle t]$

Table 2. Maximum errors for Test Problem 2.

For Test Problem 2, we use $\triangle t=1,0.2,0.05$ . Numerical calculation is executed for theinterval $[0,1]$ . Here, we use the same order approximation in space and time for simplicity.
Table 2 show that the results for small
are mostly better than the results for large
inprecision. Here, we can not adopt large $N_{x}=N_{t}$ due to the restriction on the computing$\triangle t$

resources.

$\triangle t$

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4

Conclusion

We considered a method for numerical simulation in arbitrary precision. It consists of thespectral collocation method and multiple precision arithmetic. One-phase stefan problems
whose exact solution are known are solved by this method. The numerical results are satisfactory in accuracy. Numerical simulation of one-phase stefan problems in arbitrary precision
is possible by using the spectral collocation method and multiple precision arithmetic.

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$\mathrm{N}\mathrm{e}\mathrm{w}\mathrm{Y}_{0}\mathrm{r}\mathrm{k}$[2] David, M. Smith. ”A FORTRAN Package For Floating-Point Multiple-Precision Arithmetic”.

Transactions on Mathematical Software. 17,$\mathrm{p}\mathrm{p}.273- 283(1991)$.

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Equations in Infinite Precision, Advances in Mathematical Sciences and Applications, Vol.9,
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