Three-stage Hermite–Birkhoff solver of order 8 and 9 with variable step size for stiff ODEs

Abstract

Variable-step (VS) $3$-stage Hermite–Birkhoff (HB) methods HB$(p=k+1)$ of order $p=8$ and 9 are constructed as a combination of linear $k$-step methods of order $(p-2)$ and a diagonally implicit one-step $3$-stage Runge–Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge–Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop A-stable methods of order up to 5 and A($\alpha$)-stable methods of order up to 10. Fast algorithms are developed for solving these systems in O$(p^2)$ operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsize of these methods are controlled by a local error estimator. When programmed in C++, HB($p$) of order $p=8$ and 9 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7–8), in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.

Appendices

Appendix A: Algorithms

Algorithm 1

This algorithm constructs $L_k(i,i-1)$ entries of lower bidiagonal matrices $L_k$ (applied to ${{\mathrm{\hbox {IF}}}}$, ${{\mathrm{\hbox {P}}}}_2$ and ${{\mathrm{\hbox {P}}}}_3$) as functions of $\eta _j$, $j=2,3,\ldots ,p-1$.

• For $k=2:k_{\text {end}}$, do the following iteration:

  • For $i=i_0:-1:k+1$, do the following two steps:

    • Step (1)   $L_k(i,i-1)=-M^\ell (i,k)/M^\ell (i-1,k)$.

    • Step (2)   For $j=k:m$, compute:

      $$\begin{aligned} M^\ell (i,j) = M^\ell (i,j)+M^\ell (i-1,j)L_k(i,i-1), \end{aligned}$$

  where $k_{\text {end}} =m-2, i_{0}= m$ for ${{\mathrm{\hbox {IF}}}}$, $k_{\text {end}} =m-1, i_{0}= m$ for ${{\mathrm{\hbox {P}}}}_2$ and $k_{\text {end}} =m-2, i_{0}= m-1$ for ${{\mathrm{\hbox {P}}}}_3$.

Algorithm 2

This algorithm constructs diagonal entries $U_k(j,j)$ of upper bidiagonal matrices $U_k$ applied to ${{\mathrm{\hbox {IF}}}}$ as functions of $\eta _j$, $j=2,3,\ldots ,p-1$.

• For $k=1:m_1-3$, do the following iteration:

  • For $j=m_1-2:-1:k+1$, do the following two steps:

    • Step (1)   $U_k(j,j)=1/[M^1(k+1,j)-M^1(k+1,j-1)]$.

    • Step (2)   for $i=k:j$, compute

      $$\begin{aligned} M^1(i,j) = (M^1(i,j)-M^1(i,j-1))U_k(j,j). \end{aligned}$$

Algorithm 3

This algorithm constructs diagonal entries $U_k(j,j)$ of upper bidiagonal matrices $U_k$ applied to ${{\mathrm{\hbox {P}}}}_2$ as functions of $\eta _j$, $j=2,3,\ldots ,p-1$.

• For $k=1:m_2-1$, do the following iteration:

  • For $j=m_2:-1:k+1$, do the following two steps:

    • Step (1)   $U_k(j,j)=1/[M^2(k+1,j)-M^2(k+1,j-1)]$.

    • Step (2)   for $i=k:j$, compute

      $$\begin{aligned} M^2(i,j) = (M^2(i,j)-M^2(i,j-1))U_k(j,j). \end{aligned}$$

Algorithm 4

This algorithm constructs diagonal entries $U_k(j,j)$ of upper triangular matrices $U_k$ applied to ${{\mathrm{\hbox {P}}}}_3$ as functions of $\eta _j$, $j=2,3,\ldots ,p-1$.

• For $k=1$, do the following iteration:

  • For $j=m_3-2:-1:k+1$, do the following two steps:

    • Step (1)   $U_k(j,j)=1/[M^3(k+1,j)-M^3(k+1,j-1)]$.

    • Step (2)   for $i=k:j$, compute

      $$\begin{aligned} M^3(i,j) = (M^3(i,j)-M^3(i,j-1))U_k(j,j). \end{aligned}$$

• For $k=2:m_3-2$, do the following iteration:

  • For $j=m_3-1$, do the following two steps:

    • Step (1)   $U_k(j,j)=1/[M^3(k+1,j)-M^3(k+1,j-k)]$.

    • Step (2)   for $i=k:j$, compute

      $$\begin{aligned} M^3(i,j) = (M^3(i,j)-M^3(i,j-k ))U_k(j,j). \end{aligned}$$

  • For $j=m_3-2:-1:k+1$, do the following two steps:

    • Step (1)   $U_k(j,j)=1/[M^3(k+1,j)-M^3(k+1,j-1)]$.

    • Step (2)   for $i=k:j$, compute

      $$\begin{aligned} M^3(i,j) = (M^3(i,j)-M^3(i,j-1))U_k(j,j). \end{aligned}$$

Algorithm 5

This algorithm solves the systems for ${{\mathrm{\hbox {IF}}}}$ in $O(m^2)$ operations.

Given $[\eta _2, \eta _3,\ldots , \eta _{p-1} ]$ and ${\varvec{r}}=r(1:m)$, the following algorithm overwrites ${\varvec{r}}$ with the solution ${\varvec{u}}=u(1:m)$ of the system $M{\varvec{u}}={\varvec{r}}$.

• Step (1) The following iteration overwrites ${\varvec{r}}=r(1:m)$ with $L_{m-2} L_{m-3} \cdots$ $L_2 {\varvec{r}}$:

  • for $k=2,3,\ldots , m-2$, compute

    $$\begin{aligned} r(i) =r(i)+r(i-1)L_k(i,i-1),\qquad i=m, m-1,\ldots ,k+1. \end{aligned}$$

• Step (2) This step computes the coefficients $G_{m-1}(i)$, $i=1,2,\ldots ,m$ used to form the two matrices $L_{m-1}$ and $L_{m}$ which transform $W_2^1$ into a diagonal+last-column matrix $W_1^1 = L_{m} L_{m-1} W_2^1$:

  • First set $G_{m-1}(1:m)$,

    $$\begin{aligned} G_{m-1}(1:m) = M(1:m, m-1). \end{aligned}$$

  • The following computation overwrites $G_{m-1}(1:m)$ with $L_{m-2} L_{m-3} \cdots$ $L_2 G_{m-1}(1:m)$: for $k=2,3,\ldots ,m-2$, compute

    $$\begin{aligned} G_{m-1}(i)&= G_{m-1}(i) +G_{m-1}(i-1)L_k(i,i-1),\\&\quad i=m, m-1,\ldots ,k+1. \end{aligned}$$

• Step (3) The following computation overwrites the newly obtained ${\varvec{r}}$ with $L_{m}L_{m-1}{\varvec{r}}$:

  $$\begin{aligned} r(m-1) =r(m-1)/G_{m-1}(m-1), \end{aligned}$$

  • next, for $k=m-2,m-3,\ldots ,1,$ compute

    $$\begin{aligned} r(k) =r(k)- G_{m-1}(k)r(m-1), \end{aligned}$$

  • and

    $$\begin{aligned} r(m) =r(m)- G_{m-1}(m)r(m-1). \end{aligned}$$

• Step (4) This step computes the coefficients $G_{m}(i)$, $i=1,2,\ldots ,m$ used to form the two matrices $L_{m+1}$ and $L_{m+2}$ which transform $W_1^1$ into the identity matrix $I^1 = L_{m+2} L_{m+1} W_1^1$:

  • First set $G_{m}(1:m)$:

    $$\begin{aligned} G_{m}(1:m) = M(1:m, m). \end{aligned}$$

  • The following computation overwrites $G_{m}(1:m)$ with $L_{m-2} L_{m-3} \cdots L_2 G_{m}(1:m)$: for $k=2,3,\ldots ,m-2$, compute

    $$\begin{aligned} G_{m}(i)&= G_{m}(i) +G_{m}(i-1)L_k(i,i-1),\\&\quad i=m, m-1,\ldots ,k+1. \end{aligned}$$

  • The following computation overwrites the newly obtained $G_{m}(1:m)$ with $L_{m} L_{m-1} G_{m}(1:m)$:

    $$\begin{aligned} G_{m}(m-1) =G_{m}(m-1)/G_{m-1}(m-1), \end{aligned}$$

  next, for $k=m-2,m-3,\ldots ,1,$ compute

  $$\begin{aligned} G_{m}(k) =G_{m}(k)- G_{m-1}(k) G_{m}(m-1), \end{aligned}$$

  and

  $$\begin{aligned} G_{m}(m) =G_{m}(m)- G_{m-1}(m)G_{m}(m-1). \end{aligned}$$

• Step (5) The following computation overwrites the newly obtained ${\varvec{r}}$ with $L_{m+2}L_{m+1}{\varvec{r}}$:

  $$\begin{aligned} r(m) =r(m)/G_{m}(m-1), \end{aligned}$$

  • next, for $k=m-1,m-2,\ldots ,1,$ compute

    $$\begin{aligned} r(k) =r(k)- G_{m}(k)r(m). \end{aligned}$$

• Step (6) The following iteration overwrites ${\varvec{r}}=r(1:m)$

  • with $U_1 U_2 \cdots U_{m-3} {\varvec{r}}$: For $k=m-3, m-4,\ldots ,1,$ compute

    $$\begin{aligned} r(i)&= r(i)U_k(i,i),\quad i=k+1,k+2,\ldots , m-2, \\ r(i)&= r(i)-r(i+1),\quad i=k,k+1,\ldots ,m-3. \end{aligned}$$

Algorithm 6

This algorithm solves the systems for ${{\mathrm{\hbox {P}}}}_2$ in $O(m^2)$ operations.

Given $[\eta _2, \eta _3,\ldots , \eta _{p-1} ]$ and ${\varvec{r}}=r(1:m)$, the following algorithm overwrites ${\varvec{r}}$ with the solution ${\varvec{u}}=u(1:m)$ of the system $M{\varvec{u}}={\varvec{r}}$.

• Step (1) The following iteration overwrites ${\varvec{r}}=r(1:m)$ with $L_{m-1} L_{m-2} \cdots L_2 {\varvec{r}}$:

  • for $k=2,3,\ldots , m-1$, compute

    $$\begin{aligned} r(i) =r(i)+r(i-1)L_k(i,i-1),\quad i=m, m-1,\ldots ,k+1. \end{aligned}$$

• Step (2) The following iteration overwrites ${\varvec{r}}=r(1:m)$

  • with $U_1 U_2 \cdots U_{m-1} {\varvec{r}}$: For $k=m-1, m-2,\ldots ,1,$ compute

    $$\begin{aligned} r(i)&= r(i)U_k(i,i),\quad i=k+1,k+2,\ldots , m, \\ r(i)&= r(i)-r(i+1), \quad i=k,k+1,\ldots ,m-1. \end{aligned}$$

Algorithm 7

This algorithm solves the systems for ${{\mathrm{\hbox {P}}}}_3$ in $O(m^2)$ operations.

Given $[\eta _2, \eta _3,\ldots , \eta _{p-1} ]$ and ${\varvec{r}}=r(1:m)$, the following algorithm overwrites ${\varvec{r}}$ with the solution ${\varvec{u}}=u(1:m)$ of the system $M{\varvec{u}}={\varvec{r}}$.

• Step (1) The following iteration overwrites ${\varvec{r}}=r(1:m)$ with $L_{m-2} L_{m-3} \cdots L_2 {\varvec{r}}$:

  • for $k=2,3,\ldots , m-2$, compute

    $$\begin{aligned} r(i) =r(i)+r(i-1)L_k(i,i-1),\quad i=m-1,m-2,\ldots ,k+1. \end{aligned}$$

• Step (2) This step computes the coefficients $H_{m-1}$ used to form the matrix $L_{m-1}$ which transforms $W_1^3$ into a diagonal+last-column matrix $W^3 = L_{m-1} W_1^3$:

  • First set $H_{m-1}$,

    $$\begin{aligned} H_{m-1} = M(m, m-1). \end{aligned}$$

    The following computation overwrites $H_{m-1} = M(m,m-1)$ with $(M(m,1:m)U_1 U_2 \ldots U_{m-2})(m,m-1)$: for $k=2,3,\ldots , m-2$, compute

    $$\begin{aligned} H_{m-1}= H_{m-1}U_k(m-1,m-1) \end{aligned}$$

• Step (3) The following computation overwrites the newly obtained ${\varvec{r}}$ with $L_{m-1}{\varvec{r}}$: compute

  $$\begin{aligned} r(m) =r(m)- H_{m-1} r(m-1). \end{aligned}$$

• Step (4) This step computes the coefficients $G_{m}(i)$, $i=1,2,\ldots ,m$ used to form the two matrices $L_{m}$ and $L_{m+1}$ which transform $W^3$ into the identity matrix $I^3 = L_{m+1} L_{m} W^3$:

  • First set $G_{m}(1:m)$:

    $$\begin{aligned} G_{m}(1:m) = M(1:m, m). \end{aligned}$$

    The following computation overwrites $G_{m}(1:m)$ with $\quad L_{m-2} L_{m-3} \cdots L_2 G_{m}(1:m)$: for $k=2,4,\ldots ,m-2$, compute

    $$\begin{aligned} G_{m}(i)&= G_{m}(i) +G_{m}(i-1)L_k(i,i-1),\\&\quad i=m-1, m-2,\ldots ,k+1. \end{aligned}$$

    The following computation overwrites the newly obtained $G_{m}(1:m)$ with $L_{m-1} G_{m}(1:m)$: compute

    $$\begin{aligned} G_{m}(m ) =G_{m}(m )-H_{m-1} G_{m}( m-1 ). \end{aligned}$$

• Step (5) The following computation overwrites the newly obtained ${\varvec{r}}$ with $L_{m+1}L_{m}{\varvec{r}}$:

  $$\begin{aligned} r(m) =r(m)/G_{m}(m), \end{aligned}$$

  • next, compute

    $$\begin{aligned} r(k) =r(k)- G_{m}(k)r(m),\quad k=m-1,m-2,\ldots ,1. \end{aligned}$$

• Step (6) The following iteration overwrites the newly obtained ${\varvec{r}}=r(1:m)$

  • with $U_{m-2} {\varvec{r}}$:

  • For $k=m-2$ compute

    $$\begin{aligned} r(i)&= r(i)U_k(i,i),\quad i=k+1,k+2,\ldots , m-1, \\ r(i)&= r(i)-r(i+1), \quad i=k,k+1,\ldots ,m-2. \end{aligned}$$

• Step (7) The following iteration overwrites the newly obtained ${\varvec{r}}=r(1:m)$

  • with $U_2 U_3 \cdots U_{m-3} {\varvec{r}}$:

  • For $k=m-3, m-4,\ldots ,2,$ compute

    $$\begin{aligned} r(i)&= r(i)U_k(i,i),\quad i=k+1,k+2,\ldots , m-1, \\ r(k)&= r(k)-r(k+1)-r(m-1), \\ r(i)&= r(i)-r(i+1), \quad i=k+1,k+2,\ldots ,m-3. \end{aligned}$$

• Step (8) The following iteration overwrites the newly obtained ${\varvec{r}}=r(1:m)$ with $U_1 {\varvec{r}}$:

  • For $k=1,$ compute

    $$\begin{aligned} r(i)&= r(i)U_k(i,i),\quad i=k+1,k+2,\ldots , m-2, \\ r(i)&= r(i)-r(i+1),\quad i=k,k+1,\ldots ,m-3. \end{aligned}$$

Appendix B: Coefficients of DIRK3 and HB$(p)$, $p=3,4, \ldots ,10$

The appendix lists the coefficients of DIRK3 and HB$(p)$, of order $p=3,4, \ldots ,10$, considered in this paper. It is to be noted that, in Table 4, 5 and 6, since $a_{22}=a_{33}=b_4$, only $b_4$ are listed.

Table 4 Coefficients of the implicit predictors ${{\mathrm{\hbox {P}}}}_i$, $i=2,3$ and of the integration formulae of DIRK3 and HB($p$), $p=3,4$

Table 5 Coefficients of the implicit predictors ${{\mathrm{\hbox {P}}}}_i$, $i=2,3$ and of the integration formulae of HB($p$), $p=5,6,7$

Table 6 Coefficients of the implicit predictors ${{\mathrm{\hbox {P}}}}_i$, $i=2,3$ and of the integration formulae of HB($p$), $p=8,9,10$