# Appendix: Butcher tables¶

Here we catalog the full set of Butcher tables included in ARKode. We group these into three categories: explicit, implicit and additive. However, since the methods that comprise an additive Runge Kutta method are themselves explicit and implicit, their component Butcher tables are listed within their separate sections, but are referenced together in the additive section.

In each of the following tables, we use the following notation (shown for a 3-stage method):

$\begin{split}\begin{array}{r|ccc} c_1 & a_{1,1} & a_{1,2} & a_{1,3} \\ c_2 & a_{2,1} & a_{2,2} & a_{2,3} \\ c_3 & a_{3,1} & a_{3,2} & a_{3,3} \\ \hline q & b_1 & b_2 & b_3 \\ p & \tilde{b}_1 & \tilde{b}_2 & \tilde{b}_3 \end{array}\end{split}$

where here the method and embedding share stage $$A$$ and $$c$$ values, but use their stages $$z_i$$ differently through the coefficients $$b$$ and $$\tilde{b}$$ to generate methods of orders $$q$$ (the main method) and $$p$$ (the embedding, typically $$q = p+1$$, though sometimes this is reversed).

Method authors often use different naming conventions to categorize their methods. For each of the methods below with an embedding, we follow the uniform naming convention:

NAME-S-P-Q

where here

• NAME is the author or the name provided by the author (if applicable),
• S is the number of stages in the method,
• P is the global order of accuracy for the embedding,
• Q is the global order of accuracy for the method.

For methods without an embedding (e.g., fixed-step methods) P is omitted so that methods follow the naming convention NAME-S-Q.

In the code, unique integer IDs are defined inside arkode_butcher_erk.h and arkode_butcher_dirk.h for each method, which may be used by calling routines to specify the desired method. These names are specified in fixed width font at the start of each method’s section below.

Additionally, for each method we provide a plot of the linear stability region in the complex plane. These have been computed via the following approach. For any Runge Kutta method as defined above, we may define the stability function

$R(\eta) = 1 + \eta b [I - \eta A]^{-1} e,$

where $$e\in\mathbb{R}^s$$ is a column vector of all ones, $$\eta = h\lambda$$ and $$h$$ is the time step size. If the stability function satisfies $$|R(\eta)| \le 1$$ for all eigenvalues, $$\lambda$$, of $$\frac{\partial }{\partial y}f(t,y)$$ for a given IVP, then the method will be linearly stable for that problem and step size. The stability region

$S = \{ \eta\in\mathbb{C}\; :\; \left| R(\eta) \right| \le 1\}$

is typically given by an enclosed region of the complex plane, so it is standard to search for the border of that region in order to understand the method. Since all complex numbers with unit magnitude may be written as $$e^{i\theta}$$ for some value of $$\theta$$, we perform the following algorithm to trace out this boundary.

1. Define an array of values Theta. Since we wish for a smooth curve, and since we wish to trace out the entire boundary, we choose 10,000 linearly-spaced points from 0 to $$16\pi$$. Since some angles will correspond to multiple locations on the stability boundary, by going beyond $$2\pi$$ we ensure that all boundary locations are plotted, and by using such a fine discretization the Newton method (next step) is more likely to converge to the root closest to the previous boundary point, ensuring a smooth plot.

2. For each value $$\theta \in$$ Theta, we solve the nonlinear equation

$0 = f(\eta) = R(\eta) - e^{i\theta}$

using a finite-difference Newton iteration, using tolerance $$10^{-7}$$, and differencing parameter $$\sqrt{\varepsilon}$$ ($$\approx 10^{-8}$$).

In this iteration, we use as initial guess the solution from the previous value of $$\theta$$, starting with an initial-initial guess of $$\eta=0$$ for $$\theta=0$$.

3. We then plot the resulting $$\eta$$ values that trace the stability region boundary.

We note that for any stable IVP method, the value $$\eta_0 = -\varepsilon + 0i$$ is always within the stability region. So in each of the following pictures, the interior of the stability region is the connected region that includes $$\eta_0$$. Resultingly, methods whose linear stability boundary is located entirely in the right half-plane indicate an A-stable method.

## Explicit Butcher tables¶

In the category of explicit Runge-Kutta methods, ARKode includes methods that have orders 2 through 6, with embeddings that are of orders 1 through 5.

### Heun-Euler-2-1-2¶

Accessible via the constant HEUN_EULER_2_1_2 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK(). This is the default 2nd order explicit method.

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cc} 0 & 0 & 0 \\ 1 & 1 & 0 \\ \hline 2 & \frac{1}{2} & \frac{1}{2} \\ 1 & 1 & 0 \end{array}\end{split}$

Linear stability region for the Heun-Euler method. The method’s region is outlined in blue; the embedding’s region is in red.

### Bogacki-Shampine-4-2-3¶

Accessible via the constant BOGACKI_SHAMPINE_4_2_3 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK(). This is the default 3rd order explicit method (from [BS1989]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccc} 0 & 0 & 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\ \frac{3}{4} & 0 & \frac{3}{4} & 0 & 0 \\ 1 & \frac{2}{9} & \frac{1}{3} & \frac{4}{9} & 0 \\ \hline 3 & \frac{2}{9} & \frac{1}{3} & \frac{4}{9} \\ 2 & \frac{7}{24} & \frac{1}{4} & \frac{1}{3} & \frac{1}{8} \end{array}\end{split}$

Linear stability region for the Bogacki-Shampine method. The method’s region is outlined in blue; the embedding’s region is in red.

### ARK-4-2-3 (explicit)¶

Accessible via the constant ARK324L2SA_ERK_4_2_3 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK(). This is the explicit portion of the default 3rd order additive method (from [KC2003]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccc} 0 & 0 & 0 & 0 & 0 \\ \frac{1767732205903}{2027836641118} & \frac{1767732205903}{2027836641118} & 0 & 0 & 0 \\ \frac{3}{5} & \frac{5535828885825}{10492691773637} & \frac{788022342437}{10882634858940} & 0 & 0 \\ 1 & \frac{6485989280629}{16251701735622} & -\frac{4246266847089}{9704473918619} & \frac{10755448449292}{10357097424841} & 0 \\ \hline 3 & \frac{1471266399579}{7840856788654} & -\frac{4482444167858}{7529755066697} & \frac{11266239266428}{11593286722821} & \frac{1767732205903}{4055673282236} \\ 2 & \frac{2756255671327}{12835298489170} & -\frac{10771552573575}{22201958757719} & \frac{9247589265047}{10645013368117} & \frac{2193209047091}{5459859503100} \end{array}\end{split}$

Linear stability region for the explicit ARK-4-2-3 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Knoth-Wolke-3-3¶

Accessible via the constant KNOTH_WOLKE_3_3 to MRIStepSetMRITableNum() and ARKodeLoadButcherTable_ERK(). This is the default 3th order slow and fast MRIStep method (from [KW1998]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccc} 0 & 0 & 0 & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 & 0 \\ \frac{3}{4} & -\frac{3}{16} & \frac{15}{16} & 0 \\ \hline 3 & \frac{1}{6} & \frac{3}{10} & \frac{8}{15} \end{array}\end{split}$

Linear stability region for the Knoth-Wolke method

### Zonneveld-5-3-4¶

Accessible via the constant ZONNEVELD_5_3_4 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK(). This is the default 4th order explicit method (from [Z1963]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ \frac{3}{4} & \frac{5}{32} & \frac{7}{32} & \frac{13}{32} & -\frac{1}{32} & 0 \\ \hline 4 & \frac{1}{6} & \frac{1}{3} & \frac{1}{3} & \frac{1}{6} & 0 \\ 3 & -\frac{1}{2} & \frac{7}{3} & \frac{7}{3} & \frac{13}{6} & -\frac{16}{3} \end{array}\end{split}$

Linear stability region for the Zonneveld method. The method’s region is outlined in blue; the embedding’s region is in red.

### ARK-6-3-4 (explicit)¶

Accessible via the constant ARK436L2SA_ERK_6_3_4 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK(). This is the explicit portion of the default 4th order additive method (from [KC2003]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac12 & \frac12 & 0 & 0 & 0 & 0 & 0 \\ \frac{83}{250} & \frac{13861}{62500} & \frac{6889}{62500} & 0 & 0 & 0 & 0 \\ \frac{31}{50} & -\frac{116923316275}{2393684061468} & -\frac{2731218467317}{15368042101831} & \frac{9408046702089}{11113171139209} & 0 & 0 & 0 \\ \frac{17}{20} & -\frac{451086348788}{2902428689909} & -\frac{2682348792572}{7519795681897} & \frac{12662868775082}{11960479115383} & \frac{3355817975965}{11060851509271} & 0 & 0 \\ 1 & \frac{647845179188}{3216320057751} & \frac{73281519250}{8382639484533} & \frac{552539513391}{3454668386233} & \frac{3354512671639}{8306763924573} & \frac{4040}{17871} & 0 \\ \hline 4 & \frac{82889}{524892} & 0 & \frac{15625}{83664} & \frac{69875}{102672} & -\frac{2260}{8211} & \frac14 \\ 3 & \frac{4586570599}{29645900160} & 0 & \frac{178811875}{945068544} & \frac{814220225}{1159782912} & -\frac{3700637}{11593932} & \frac{61727}{225920} \end{array}\end{split}$

Linear stability region for the explicit ARK-6-3-4 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Sayfy-Aburub-6-3-4¶

Accessible via the constant SAYFY_ABURUB_6_3_4 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK() (from [SA2002]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 2 & 0 & 0 & 0 & 0 \\ 1 & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} & 0 & 0 & 0 \\ \frac{1}{2} & 0.137 & 0.226 & 0.137 & 0 & 0 & 0 \\ 1 & 0.452 & -0.904 & -0.548 & 0 & 2 & 0 \\ \hline 4 & \frac{1}{6} & \frac{1}{3} & \frac{1}{12} & 0 & \frac{1}{3} & \frac{1}{12} \\ 3 & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} & 0 & 0 & 0 \end{array}\end{split}$

Linear stability region for the Sayfy-Aburub-6-3-4 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Cash-Karp-6-4-5¶

Accessible via the constant CASH_KARP_6_4_5 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK(). This is the default 5th order explicit method (from [CK1990]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 0 & 0 \\ \frac{3}{10} & \frac{3}{40} & \frac{9}{40} & 0 & 0 & 0 & 0 \\ \frac{3}{5} & \frac{3}{10} & -\frac{9}{10} & \frac{6}{5} & 0 & 0 & 0 \\ 1 & -\frac{11}{54} & \frac{5}{2} & -\frac{70}{27} & \frac{35}{27} & 0 & 0 \\ \frac{7}{8} & \frac{1631}{55296} & \frac{175}{512} & \frac{575}{13824} & \frac{44275}{110592} & \frac{253}{4096} & 0 \\ \hline 5 & \frac{37}{378} & 0 & \frac{250}{621} & \frac{125}{594} & 0 & \frac{512}{1771} \\ 4 & \frac{2825}{27648} & 0 & \frac{18575}{48384} & \frac{13525}{55296} & \frac{277}{14336} & \frac{1}{4} \end{array}\end{split}$

Linear stability region for the Cash-Karp method. The method’s region is outlined in blue; the embedding’s region is in red.

### Fehlberg-6-4-5¶

Accessible via the constant FEHLBERG_6_4_5 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK() (from [F1969]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{4} & \frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ \frac{3}{8} & \frac{3}{32} & \frac{9}{32} & 0 & 0 & 0 & 0 \\ \frac{12}{13} & \frac{1932}{2197} & -\frac{7200}{2197} & \frac{7296}{2197} & 0 & 0 & 0 \\ 1 & \frac{439}{216} & -8 & \frac{3680}{513} & -\frac{845}{4104} & 0 & 0 \\ \frac{1}{2} & -\frac{8}{27} & 2 & -\frac{3544}{2565} & \frac{1859}{4104} & -\frac{11}{40} & 0 \\ \hline 5 & \frac{16}{135} & 0 & \frac{6656}{12825} & \frac{28561}{56430} & -\frac{9}{50} & \frac{2}{55} \\ 4 & \frac{25}{216} & 0 & \frac{1408}{2565} & \frac{2197}{4104} & -\frac{1}{5} & 0 \end{array}\end{split}$

Linear stability region for the Fehlberg method. The method’s region is outlined in blue; the embedding’s region is in red.

### Dormand-Prince-7-4-5¶

Accessible via the constant DORMAND_PRINCE_7_4_5 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK() (from [DP1980]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{5} & \frac{1}{5} & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{3}{10} & \frac{3}{40} & \frac{9}{40} & 0 & 0 & 0 & 0 & 0 \\ \frac{4}{5} & \frac{44}{45} & -\frac{56}{15} & \frac{32}{9} & 0 & 0 & 0 & 0 \\ \frac{8}{9} & \frac{19372}{6561} & -\frac{25360}{2187} & \frac{64448}{6561} & -\frac{212}{729} & 0 & 0 & 0 \\ 1 & \frac{9017}{3168} & -\frac{355}{33} & \frac{46732}{5247} & \frac{49}{176} & -\frac{5103}{18656} & 0 & 0 \\ 1 & \frac{35}{384} & 0 & \frac{500}{1113} & \frac{125}{192} & -\frac{2187}{6784} & \frac{11}{84} & 0 \\ \hline 5 & \frac{35}{384} & 0 & \frac{500}{1113} & \frac{125}{192} & -\frac{2187}{6784} & \frac{11}{84} & 0 \\ 4 & \frac{5179}{57600} & 0 & \frac{7571}{16695} & \frac{393}{640} & -\frac{92097}{339200} & \frac{187}{2100} & \frac{1}{40} \end{array}\end{split}$

Linear stability region for the Dormand-Prince method. The method’s region is outlined in blue; the embedding’s region is in red.

### ARK-8-4-5 (explicit)¶

Accessible via the constant ARK548L2SA_ERK_8_4_5 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK(). This is the explicit portion of the default 5th order additive method (from [KC2003]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{41}{100} & \frac{41}{100} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{2935347310677}{11292855782101} & \frac{367902744464}{2072280473677} & \frac{677623207551}{8224143866563} & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1426016391358}{7196633302097} & \frac{1268023523408}{10340822734521} & 0 & \frac{1029933939417}{13636558850479} & 0 & 0 & 0 & 0 & 0 \\ \frac{92}{100} & \frac{14463281900351}{6315353703477} & 0 & \frac{66114435211212}{5879490589093} & -\frac{54053170152839}{4284798021562} & 0 & 0 & 0 & 0 \\ \frac{24}{100} & \frac{14090043504691}{34967701212078} & 0 & \frac{15191511035443}{11219624916014} & -\frac{18461159152457}{12425892160975} & -\frac{281667163811}{9011619295870} & 0 & 0 & 0 \\ \frac{3}{5} & \frac{19230459214898}{13134317526959} & 0 & \frac{21275331358303}{2942455364971} & -\frac{38145345988419}{4862620318723} & -\frac{1}{8} & -\frac{1}{8} & 0 & 0 \\ 1 & -\frac{19977161125411}{11928030595625} & 0 & -\frac{40795976796054}{6384907823539} & \frac{177454434618887}{12078138498510} & \frac{782672205425}{8267701900261} & -\frac{69563011059811}{9646580694205} & \frac{7356628210526}{4942186776405} & 0 \\ \hline 5 & -\frac{872700587467}{9133579230613} & 0 & 0 & \frac{22348218063261}{9555858737531} & -\frac{1143369518992}{8141816002931} & -\frac{39379526789629}{19018526304540} & \frac{32727382324388}{42900044865799} & \frac{41}{200} \\ 4 & -\frac{975461918565}{9796059967033} & 0 & 0 & \frac{78070527104295}{32432590147079} & -\frac{548382580838}{3424219808633} & -\frac{33438840321285}{15594753105479} & \frac{3629800801594}{4656183773603} & \frac{4035322873751}{18575991585200} \end{array}\end{split}$

Linear stability region for the explicit ARK-8-4-5 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Verner-8-5-6¶

Accessible via the constant VERNER_8_5_6 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK(). This is the default 6th order explicit method (from [V1978]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{6} & \frac{1}{6} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{4}{15} & \frac{4}{75} & \frac{16}{75} & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{2}{3} & \frac{5}{6} & -\frac{8}{3} & \frac{5}{2} & 0 & 0 & 0 & 0 & 0 \\ \frac{5}{6} & -\frac{165}{64} & \frac{55}{6} & -\frac{425}{64} & \frac{85}{96} & 0 & 0 & 0 & 0 \\ 1 & \frac{12}{5} & -8 & \frac{4015}{612} & -\frac{11}{36} & \frac{88}{255} & 0 & 0 & 0 \\ \frac{1}{15} & -\frac{8263}{15000} & \frac{124}{75} & -\frac{643}{680} & -\frac{81}{250} & \frac{2484}{10625} & 0 & 0 & 0 \\ 1 & \frac{3501}{1720} & -\frac{300}{43} & \frac{297275}{52632} & -\frac{319}{2322} & \frac{24068}{84065} & 0 & \frac{3850}{26703} & 0 \\ \hline 6 & \frac{3}{40} & 0 & \frac{875}{2244} & \frac{23}{72} & \frac{264}{1955} & 0 & \frac{125}{11592} & \frac{43}{616} \\ 5 & \frac{13}{160} & 0 & \frac{2375}{5984} & \frac{5}{16} & \frac{12}{85} & \frac{3}{44} & 0 & 0 \end{array}\end{split}$

Linear stability region for the Verner-8-5-6 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Fehlberg-13-7-8¶

Accessible via the constant FEHLBERG_13_7_8 to ARKStepSetARKTableNum(), ERKStepSetERKTableNum() or ARKodeLoadButcherTable_ERK(). This is the default 8th order explicit method (from [B2008]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccccccccccccc} 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \frac{2}{27}& \frac{2}{27}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \frac{1}{9}& \frac{1}{36}& \frac{1}{12}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \frac{1}{6}& \frac{1}{24}& 0& \frac{1}{8}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \frac{5}{12}& \frac{5}{12}& 0& -\frac{25}{16}& \frac{25}{16}& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ \frac{1}{2}& \frac{1}{20}& 0& 0& \frac{1}{4}& \frac{1}{5}& 0& 0& 0& 0& 0& 0& 0& 0\\ \frac{5}{6}& -\frac{25}{108}& 0& 0& \frac{125}{108}& -\frac{65}{27}& \frac{125}{54}& 0& 0& 0& 0& 0& 0& 0\\ \frac{1}{6}& \frac{31}{300}& 0& 0& 0& \frac{61}{225}& -\frac{2}{9}& \frac{13}{900}& 0& 0& 0& 0& 0& 0\\ \frac{2}{3}& 2& 0& 0& -\frac{53}{6}& \frac{704}{45}& -\frac{107}{9}& \frac{67}{90}& 3& 0& 0& 0& 0& 0\\ \frac{1}{3}& -\frac{91}{108}& 0& 0& \frac{23}{108}& -\frac{976}{135}& \frac{311}{54}& -\frac{19}{60}& \frac{17}{6}& -\frac{1}{12}& 0& 0& 0& 0\\ 1& \frac{2383}{4100}& 0& 0& -\frac{341}{164}& \frac{4496}{1025}& -\frac{301}{82}& \frac{2133}{4100}& \frac{45}{82}& \frac{45}{164}& \frac{18}{41}& 0& 0& 0\\ 0& \frac{3}{205}& 0& 0& 0& 0& -\frac{6}{41}& -\frac{3}{205}& -\frac{3}{41}& \frac{3}{41}& \frac{6}{41}& 0& 0& 0\\ 1& -\frac{1777}{4100}& 0& 0& -\frac{341}{164}& \frac{4496}{1025}& -\frac{289}{82}& \frac{2193}{4100}& \frac{51}{82}& \frac{33}{164}& \frac{12}{41}& 0& 1& 0\\ \hline 8& 0& 0& 0& 0& 0& \frac{34}{105}& \frac{9}{35}& \frac{9}{35}& \frac{9}{280}& \frac{9}{280}& 0& \frac{41}{840}& \frac{41}{840} \\ 7& \frac{41}{840}& 0& 0& 0& 0& \frac{34}{105}& \frac{9}{35}& \frac{9}{35}& \frac{9}{280}& \frac{9}{280}& \frac{41}{840}& 0& 0 \end{array}\end{split}$

Linear stability region for the Fehlberg-13-7-8 method. The method’s region is outlined in blue; the embedding’s region is in red.

## Implicit Butcher tables¶

In the category of diagonally implicit Runge-Kutta methods, ARKode includes methods that have orders 2 through 5, with embeddings that are of orders 1 through 4.

### SDIRK-2-1-2¶

Accessible via the constant SDIRK_2_1_2 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). This is the default 2nd order implicit method. Both the method and embedding are A- and B-stable.

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cc} 1 & 1 & 0 \\ 0 & -1 & 1 \\ \hline 2 & \frac{1}{2} & \frac{1}{2} \\ 1 & 1 & 0 \end{array}\end{split}$

Linear stability region for the SDIRK-2-1-2 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Billington-3-3-2¶

Accessible via the constant BILLINGTON_3_3_2 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). Here, the higher-order embedding is less stable than the lower-order method (from [B1983]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccc} 0.292893218813 & 0.292893218813 & 0 & 0 \\ 1.091883092037 & 0.798989873223 & 0.292893218813 & 0 \\ 1.292893218813 & 0.740789228841 & 0.259210771159 & 0.292893218813 \\ \hline 2 & 0.740789228840 & 0.259210771159 & 0 \\ 3 & 0.691665115992 & 0.503597029883 & -0.195262145876 \end{array}\end{split}$

Linear stability region for the Billington method. The method’s region is outlined in blue; the embedding’s region is in red.

### TRBDF2-3-3-2¶

Accessible via the constant TRBDF2_3_3_2 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). As with Billington, here the higher-order embedding is less stable than the lower-order method (from [B1985]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccc} 0 & 0 & 0 & 0 \\ 2-\sqrt{2} & \frac{2-\sqrt{2}}{2} & \frac{2-\sqrt{2}}{2} & 0 \\ 1 & \frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{4} & \frac{2-\sqrt{2}}{2} \\ \hline 2 & \frac{\sqrt{2}}{4} & \frac{\sqrt{2}}{4} & \frac{2-\sqrt{2}}{2} \\ 3 & \frac{1-\frac{\sqrt{2}}{4}}{3} & \frac{\frac{3\sqrt{2}}{4}+1}{3} & \frac{2-\sqrt{2}}{6} \end{array}\end{split}$

Linear stability region for the TRBDF2 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Kvaerno-4-2-3¶

Accessible via the constant KVAERNO_4_2_3 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). Both the method and embedding are A-stable; additionally the method is L-stable (from [K2004]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccc} 0 & 0 & 0 & 0 & 0 \\ 0.871733043 & 0.4358665215 & 0.4358665215 & 0 & 0 \\ 1 & 0.490563388419108 & 0.073570090080892 & 0.4358665215 & 0 \\ 1 & 0.308809969973036 & 1.490563388254106 & -1.235239879727145 & 0.4358665215 \\ \hline 3 & 0.308809969973036 & 1.490563388254106 & -1.235239879727145 & 0.4358665215 \\ 2 & 0.490563388419108 & 0.073570090080892 & 0.4358665215 & 0 \end{array}\end{split}$

Linear stability region for the Kvaerno-4-2-3 method. The method’s region is outlined in blue; the embedding’s region is in red.

### ARK-4-2-3 (implicit)¶

Accessible via the constant ARK324L2SA_DIRK_4_2_3 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). This is the default 3rd order implicit method, and the implicit portion of the default 3rd order additive method. Both the method and embedding are A-stable; additionally the method is L-stable (from [KC2003]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccc} 0 & 0 & 0 & 0 & 0 \\ \frac{1767732205903}{2027836641118} & \frac{1767732205903}{4055673282236} & \frac{1767732205903}{4055673282236} & 0 & 0 \\ \frac{3}{5} & \frac{2746238789719}{10658868560708} & -\frac{640167445237}{6845629431997} & \frac{1767732205903}{4055673282236} & 0 \\ 1 & \frac{1471266399579}{7840856788654} & -\frac{4482444167858}{7529755066697} & \frac{11266239266428}{11593286722821} & \frac{1767732205903}{4055673282236} \\ \hline 3 & \frac{1471266399579}{7840856788654} & -\frac{4482444167858}{7529755066697} & \frac{11266239266428}{11593286722821} & \frac{1767732205903}{4055673282236} \\ 2 & \frac{2756255671327}{12835298489170} & -\frac{10771552573575}{22201958757719} & \frac{9247589265047}{10645013368117} & \frac{2193209047091}{5459859503100} \end{array}\end{split}$

Linear stability region for the implicit ARK-4-2-3 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Cash-5-2-4¶

Accessible via the constant CASH_5_2_4 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). Both the method and embedding are A-stable; additionally the method is L-stable (from [C1979]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccccc} 0.435866521508 & 0.435866521508 & 0 & 0 & 0 & 0 \\ -0.7 & -1.13586652150 & 0.435866521508 & 0 & 0 & 0 \\ 0.8 & 1.08543330679 & -0.721299828287 & 0.435866521508 & 0 & 0 \\ 0.924556761814 & 0.416349501547 & 0.190984004184 & -0.118643265417 & 0.435866521508 & 0 \\ 1 & 0.896869652944 & 0.0182725272734 & -0.0845900310706 & -0.266418670647 & 0.435866521508 \\ \hline 4 & 0.896869652944 & 0.0182725272734 & -0.0845900310706 & -0.266418670647 & 0.435866521508 \\ 2 & 1.05646216107052 & -0.0564621610705236 & 0 & 0 & 0 \end{array}\end{split}$

Linear stability region for the Cash-5-2-4 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Cash-5-3-4¶

Accessible via the constant CASH_5_3_4 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). Both the method and embedding are A-stable; additionally the method is L-stable (from [C1979]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccccc} 0.435866521508 & 0.435866521508 & 0 & 0 & 0 & 0 \\ -0.7 & -1.13586652150 & 0.435866521508 & 0 & 0 & 0 \\ 0.8 & 1.08543330679 & -0.721299828287 & 0.435866521508 & 0 & 0 \\ 0.924556761814 & 0.416349501547 & 0.190984004184 & -0.118643265417 & 0.435866521508 & 0 \\ 1 & 0.896869652944 & 0.0182725272734 & -0.0845900310706 & -0.266418670647 & 0.435866521508 \\ \hline 4 & 0.896869652944 & 0.0182725272734 & -0.0845900310706 & -0.266418670647 & 0.435866521508 \\ 3 & 0.776691932910 & 0.0297472791484 & -0.0267440239074 & 0.220304811849 & 0 \end{array}\end{split}$

Linear stability region for the Cash-5-3-4 method. The method’s region is outlined in blue; the embedding’s region is in red.

### SDIRK-5-3-4¶

Accessible via the constant SDIRK_5_3_4 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). This is the default 4th order implicit method. Here, the method is both A- and L-stable, although the embedding has reduced stability (from [HW1996]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccccc} \frac{1}{4} & \frac{1}{4} & 0 & 0 & 0 & 0 \\ \frac{3}{4} & \frac{1}{2} & \frac{1}{4} & 0 & 0 & 0 \\ \frac{11}{20} & \frac{17}{50} & -\frac{1}{25} & \frac{1}{4} & 0 & 0 \\ \frac{1}{2} & \frac{371}{1360} & -\frac{137}{2720} & \frac{15}{544} & \frac{1}{4} & 0 \\ 1 & \frac{25}{24} & -\frac{49}{48} & \frac{125}{16} & -\frac{85}{12} & \frac{1}{4} \\ \hline 4 & \frac{25}{24} & -\frac{49}{48} & \frac{125}{16} & -\frac{85}{12} & \frac{1}{4} \\ 3 & \frac{59}{48} & -\frac{17}{96} & \frac{225}{32} & -\frac{85}{12} & 0 \end{array}\end{split}$

Linear stability region for the SDIRK-5-3-4 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Kvaerno-5-3-4¶

Accessible via the constant KVAERNO_5_3_4 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). Both the method and embedding are A-stable (from [K2004]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ 0.871733043 & 0.4358665215 & 0.4358665215 & 0 & 0 & 0 \\ 0.468238744853136 & 0.140737774731968 & -0.108365551378832 & 0.4358665215 & 0 & 0 \\ 1 & 0.102399400616089 & -0.376878452267324 & 0.838612530151233 & 0.4358665215 & 0 \\ 1 & 0.157024897860995 & 0.117330441357768 & 0.61667803039168 & -0.326899891110444 & 0.4358665215 \\ \hline 4 & 0.157024897860995 & 0.117330441357768 & 0.61667803039168 & -0.326899891110444 & 0.4358665215 \\ 3 & 0.102399400616089 & -0.376878452267324 & 0.838612530151233 & 0.4358665215 & 0 \end{array}\end{split}$

Linear stability region for the Kvaerno-5-3-4 method. The method’s region is outlined in blue; the embedding’s region is in red.

### ARK-6-3-4 (implicit)¶

Accessible via the constant ARK436L2SA_DIRK_6_3_4 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). This is the implicit portion of the default 4th order additive method. Both the method and embedding are A-stable; additionally the method is L-stable (from [KC2003]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{2} & \frac{1}{4} & \frac{1}{4} & 0 & 0 & 0 & 0 \\ \frac{83}{250} & \frac{8611}{62500} & -\frac{1743}{31250} & \frac{1}{4} & 0 & 0 & 0 \\ \frac{31}{50} & \frac{5012029}{34652500} & -\frac{654441}{2922500} & \frac{174375}{388108} & \frac{1}{4} & 0 & 0 \\ \frac{17}{20} & \frac{15267082809}{155376265600} & -\frac{71443401}{120774400} & \frac{730878875}{902184768} & \frac{2285395}{8070912} & \frac{1}{4} & 0 \\ 1 & \frac{82889}{524892} & 0 & \frac{15625}{83664} & \frac{69875}{102672} & -\frac{2260}{8211} & \frac{1}{4} \\ \hline 4 & \frac{82889}{524892} & 0 & \frac{15625}{83664} & \frac{69875}{102672} & -\frac{2260}{8211} & \frac{1}{4} \\ 3 & \frac{4586570599}{29645900160} & 0 & \frac{178811875}{945068544} & \frac{814220225}{1159782912} & -\frac{3700637}{11593932} & \frac{61727}{225920} \end{array}\end{split}$

Linear stability region for the implicit ARK-6-3-4 method. The method’s region is outlined in blue; the embedding’s region is in red.

### Kvaerno-7-4-5¶

Accessible via the constant KVAERNO_7_4_5 to ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). Both the method and embedding are A-stable; additionally the method is L-stable (from [K2004]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0.52 & 0.26 & 0.26 & 0 & 0 & 0 & 0 & 0 \\ 1.230333209967908 & 0.13 & 0.84033320996790809 & 0.26 & 0 & 0 & 0 & 0 \\ 0.895765984350076 & 0.22371961478320505 & 0.47675532319799699 & -0.06470895363112615 & 0.26 & 0 & 0 & 0 \\ 0.436393609858648 & 0.16648564323248321 & 0.10450018841591720 & 0.03631482272098715 & -0.13090704451073998 & 0.26 & 0 & 0 \\ 1 & 0.13855640231268224 & 0 & -0.04245337201752043 & 0.02446657898003141 & 0.61943039072480676 & 0.26 & 0 \\ 1 & 0.13659751177640291 & 0 & -0.05496908796538376 & -0.04118626728321046 & 0.62993304899016403 & 0.06962479448202728 & 0.26 \\ \hline 5 & 0.13659751177640291 & 0 & -0.05496908796538376 & -0.04118626728321046 & 0.62993304899016403 & 0.06962479448202728 & 0.26 \\ 4 & 0.13855640231268224 & 0 & -0.04245337201752043 & 0.02446657898003141 & 0.61943039072480676 & 0.26 & 0 \end{array}\end{split}$

Linear stability region for the Kvaerno-7-4-5 method. The method’s region is outlined in blue; the embedding’s region is in red.

### ARK-8-4-5 (implicit)¶

Accessible via the constant ARK548L2SA_DIRK_8_4_5 for ARKStepSetIRKTableNum() or ARKodeLoadButcherTable_DIRK(). This is the default 5th order implicit method, and the implicit portion of the default 5th order additive method. Both the method and embedding are A-stable; additionally the method is L-stable (from [KC2003]).

$\begin{split}\renewcommand{\arraystretch}{1.5} \begin{array}{r|cccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{41}{100} & \frac{41}{200} & \frac{41}{200} & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{2935347310677}{11292855782101} & \frac{41}{400} & -\frac{567603406766}{11931857230679} & \frac{41}{200} & 0 & 0 & 0 & 0 & 0 \\ \frac{1426016391358}{7196633302097} & \frac{683785636431}{9252920307686} & 0 & -\frac{110385047103}{1367015193373} & \frac{41}{200} & 0 & 0 & 0 & 0 \\ \frac{92}{100} & \frac{3016520224154}{10081342136671} & 0 & \frac{30586259806659}{12414158314087} & -\frac{22760509404356}{11113319521817} & \frac{41}{200} & 0 & 0 & 0 \\ \frac{24}{100} & \frac{218866479029}{1489978393911} & 0 & \frac{638256894668}{5436446318841} & -\frac{1179710474555}{5321154724896} & -\frac{60928119172}{8023461067671} & \frac{41}{200} & 0 & 0 \\ \frac{3}{5} & \frac{1020004230633}{5715676835656} & 0 & \frac{25762820946817}{25263940353407} & -\frac{2161375909145}{9755907335909} & -\frac{211217309593}{5846859502534} & -\frac{4269925059573}{7827059040749} & \frac{41}{200} & 0 \\ 1 & -\frac{872700587467}{9133579230613} & 0 & 0 & \frac{22348218063261}{9555858737531} & -\frac{1143369518992}{8141816002931} & -\frac{39379526789629}{19018526304540} & \frac{32727382324388}{42900044865799} & \frac{41}{200} \\ \hline 5 & -\frac{872700587467}{9133579230613} & 0 & 0 & \frac{22348218063261}{9555858737531} & -\frac{1143369518992}{8141816002931} & -\frac{39379526789629}{19018526304540} & \frac{32727382324388}{42900044865799} & \frac{41}{200} \\ 4 & -\frac{975461918565}{9796059967033} & 0 & 0 & \frac{78070527104295}{32432590147079} & -\frac{548382580838}{3424219808633} & -\frac{33438840321285}{15594753105479} & \frac{3629800801594}{4656183773603} & \frac{4035322873751}{18575991585200} \end{array}\end{split}$

Linear stability region for the implicit ARK-8-4-5 method. The method’s region is outlined in blue; the embedding’s region is in red.