One-step one-stage methods of order 2 and 3 for stiff ODEs

S.S. Filippov, M.V. Bulatov

Keldysh Institute of Applied Mathematics, 4 Miusskaya sq., 125047 Moscow, Russia
Institute for System Dynamics and Control Theory, 134 Lermontova str., 664033 Irkutsk, Russia

Filippov@Keldysh.ru, mvbul@icc.ru

Contributed talk

For the problem

$$x'(t)=f(x(t),t),\quad x(0)=x_0,\quad t\in[0,1]$$ (1)

with a sufficiently smooth function $f\in{\bf R}^n$ let us write down the explicit and the linearized implicit Euler schemes:

$$Sx_{i+1}=Sx_i+{{hf(x_i,t_i)}\choose{hf(x_i,t_{i+1})}},$$ (2)

where $S=(I\quad I-hJ)^T$ is a $2n\!\!\times\!\!n$-matrix and $J=f_x(x_i,t_{i+1})$. Genrally, (2) has no classical solution. By multiplying (2) by $S^T$ we arrive at

$$S^TSx_{i+1}=x_i+hf(x_i,t_i)+(I-hJ)\{x_i+h[f(x_i,t_{i+1})-Jx_i]\}.$$ (3)

This is a one-step one-stage method of order 2, $L$-stable, with the stability function $R(z)=1/(1-z+z^2/2)$. When using method (3) to solve numerically problem (1), we have to solve a linear system of $n$ algebraic equations only [1].

For a linear problem

$$x'(t)=B(t)x(t)+g(t),\quad x(0)=x_0,\quad t\in[0,1]$$ (4)

with $n\!\!\times\!\!n$-matrix $B(t)$ and $g(t)\in{\bf R}^n$ sufficiently smooth, one can construct a one-step one-stage method of order 3. Combining two one-step methods applied to (4) leads to the system

$$Tx_{i+1}=Ux_i+h{{pg_{i+1}+(1-p)g_i)}\choose{\nu g(t_i+hq)}},$$ (5)

of $2n$ equations for $x_{i+1}$, where $T=(I-hpB_{i+1}\quad \nu(I-hqB(t_i+hq)))^T$ and $U=(I+h(1-p)B_i\quad \nu(I+h(1-q)B(t_i+hq)))^T$ are $2n\!\!\times\!\!n$-matrices. Genrally, (5) has no classical solution. Multiplying (5) by $T^T$ and choosing $p=0$, $q=2/3$, and $\nu=\sqrt{3}$ gives the desired method:

$$\begin{array}{l} \{I+3[I-2hB(t_i+2h/3)/3]^2\}x_{i+1}= \{I+h... ...3]\}x_i+ h\{f_i+3[I-2hB(t_i+2h/3)/3]f(t_i+2h/3)\}. \end{array}$$ (6)

Again, we have to solve only a linear algebraic system of $n$ equations. The method (6) is $A$-stable with $R(z)=(4-2z^2/3)/(4-4z+4z^2/3)$.

Numerical experiments with the methods (3) and (6) were carried out for test problems taken from [2, (7.5.4)] and [3, p. 13], respectively.