2.6. SPRK Method Table Structure

To store the pair of Butcher tables defining a SPRK method of order \(q\) ARKODE provides the ARKodeSPRKTable type and several related utility routines. We use the following notation

\[\begin{split}B \; \equiv \; \begin{array}{r|c} c & A \\ \hline & b \\ \end{array} \; = \; \begin{array}{c|cccc} c_1 & 0 & \cdots & 0 & 0 \\ c_2 & a_1 & 0 & \cdots & \vdots \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ c_s & a_1 & \cdots & a_{s-1} & 0 \\ \hline & a_1 & \cdots & a_{s-1} & a_s \end{array} \qquad \qquad \hat{B} \; \equiv \; \begin{array}{r|c} \hat{c} & \hat{A} \\ \hline & \hat{b} \\ \end{array} \; = \; \begin{array}{c|cccc} \hat{c}_1 & \hat{a}_1 & \cdots & 0 & 0 \\ \hat{c}_2 & \hat{a}_1 & \hat{a}_2 & \cdots & \vdots \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ \hat{c}_s & \hat{a}_1 & \hat{a}_2 & \cdots & \hat{a}_{s} \\ \hline & \hat{a}_1 & \hat{a}_2 & \cdots & \hat{a}_{s} \end{array}\end{split}\]

where \(B\) and \(\hat{B}\) contain the coefficients for the explicit and diagonally implicit tables, respectively. We use a compact storage of these coefficients in terms of two arrays, one for \(a\) values and one for \(\hat{a}\) values. The abscissae (only relevant for non-autonomous problems) are computed dynamically as \(c_j = \sum_{i=1}^j a_i\) and \(\hat{c}_j = \sum_{i=1}^j \hat{a}_i\), respectively [44, 77]. The ARKodeSPRKTable type is a pointer to the ARKodeSPRKTableMem structure:

typedef ARKodeSPRKTableMem *ARKodeSPRKTable

type ARKodeSPRKTableMem

Structure representing the SPRK method that holds the method coefficients.

int q

The method order of accuracy.

int stages

The number of stages.

sunrealtype *a

Array of coefficients that generate the explicit Butcher table. a[i] is the coefficient appearing in column i+1.

sunrealtype *ahat

Array of coefficients that generate the diagonally-implicit Butcher table. ahat[i] is the coefficient appearing in column i.

2.6.1. ARKodeSPRKTable functions

Table 2.11 ARKodeSPRKTable functions

Function name	Description
ARKodeSPRKTable_Alloc()	Allocate an empty table
ARKodeSPRKTable_Load()	Load SPRK method using an identifier
ARKodeSPRKTable_LoadByName()	Load SPRK method using a string version of the identifier
ARKodeSPRKTable_Create()	Create a new table
ARKodeSPRKTable_Copy()	Create a copy of a table
ARKodeSPRKTable_Space()	Get the table real and integer workspace size
ARKodeSPRKTable_Free()	Deallocate a table

ARKodeSPRKTable ARKodeSPRKTable_Create(int stages, int q, const sunrealtype *a, const sunrealtype *ahat)

Creates and allocates an ARKodeSPRKTable with the specified number of stages and the coefficients provided.

Parameters:

• stages – The number of stages.

• q – The order of the method.

• a – An array of the coefficients for the a table.

• ahat – An array of the coefficients for the ahat table.

Returns:

ARKodeSPRKTable for the loaded method.

ARKodeSPRKTable ARKodeSPRKTable_Alloc(int stages)

Allocate memory for an ARKodeSPRKTable with the specified number of stages.

Parameters:

• stages – The number of stages.

Returns:

ARKodeSPRKTable for the loaded method.

ARKodeSPRKTable ARKodeSPRKTable_Load(ARKODE_SPRKMethodID id)

Load the ARKodeSPRKTable for the specified method ID.

Parameters:

• id – The ID of the SPRK method, see Symplectic Partitioned Butcher tables.

Returns:

ARKodeSPRKTable for the loaded method.

ARKodeSPRKTable ARKodeSPRKTable_LoadByName(const char *method)

Load the ARKodeSPRKTable for the specified method name.

Parameters:

• method – The name of the SPRK method, see Symplectic Partitioned Butcher tables.

Returns:

ARKodeSPRKTable for the loaded method.

ARKodeSPRKTable ARKodeSPRKTable_Copy(ARKodeSPRKTable sprk_table)

Create a copy of the ARKodeSPRKTable.

Parameters:

• sprk_table – The ARKodeSPRKTable to copy.

Returns:

Pointer to the copied ARKodeSPRKTable.

void ARKodeSPRKTable_Write(ARKodeSPRKTable sprk_table, FILE *outfile)

Write the ARKodeSPRKTable out to the file.

Parameters:

• sprk_table – The ARKodeSPRKTable to write.

• outfile – The FILE that will be written to.

void ARKodeSPRKTable_Space(ARKodeSPRKTable sprk_table, sunindextype *liw, sunindextype *lrw)

Get the workspace sizes required for the ARKodeSPRKTable.

Parameters:

• sprk_table – The ARKodeSPRKTable.

• liw – Pointer to store the integer workspace size.

• lrw – Pointer to store the real workspace size.

void ARKodeSPRKTable_Free(ARKodeSPRKTable sprk_table)

Free the memory allocated for the ARKodeSPRKTable.

Parameters:

• sprk_table – The ARKodeSPRKTable to free.

int ARKodeSPRKTable_ToButcher(ARKodeSPRKTable sprk_table, ARKodeButcherTable *a_ptr, ARKodeButcherTable *b_ptr)

Convert the ARKodeSPRKTable to the Butcher table representation.

Parameters:

• sprk_table – The ARKodeSPRKTable.

• a_ptr – Pointer to store the explicit Butcher table.

• b_ptr – Pointer to store the diagonally-implicit Butcher table.