Add DAE initialization algorithms: DefaultInit, BrownBasicInit, ShampineCollocationInit

- DefaultInit: Smart default that chooses between OverrideInit and CheckInit
- BrownBasicInit: Brown et al. (1998) method for algebraic variable initialization
- ShampineCollocationInit: Shampine (2001) collocation method for full initialization

These algorithms provide standard, well-documented methods for DAE initialization
with proper citations to the original papers.

References:
- Brown et al., SIAM J. Sci. Comput. 19(5), 1495-1512 (1998)
- Shampine, SIAM J. Sci. Comput. 22(6), 2007-2026 (2001)

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-Authored-By: Claude <[email protected]>

Author: ChrisRackauckas

src/DiffEqBase.jl

@@ -141,6 +141,7 @@ include("stats.jl")
 include("calculate_residuals.jl")
 include("tableaus.jl")
 include("internal_itp.jl")
+include("dae_initialization.jl")
 
 include("callbacks.jl")
 include("common_defaults.jl")

src/dae_initialization.jl

@@ -0,0 +1,59 @@
+# DAE Initialization Algorithms
+# The base types are in SciMLBase, we just add the specific algorithms here
+
+import SciMLBase: DAEInitializationAlgorithm, initialize_dae!
+
+"""
+    struct DefaultInit <: DAEInitializationAlgorithm
+
+The default initialization algorithm for DAEs. This will use heuristics to
+determine the most appropriate initialization based on the problem type.
+
+For Sundials, this will use:
+- `OverrideInit` if the problem has `initialization_data` (typically from ModelingToolkit)
+- `CheckInit` otherwise
+"""
+struct DefaultInit <: DAEInitializationAlgorithm end
+
+"""
+    struct BrownBasicInit <: DAEInitializationAlgorithm
+
+The Brown basic initialization algorithm for DAEs. This implementation
+is based on the algorithm described in:
+
+Peter N. Brown, Alan C. Hindmarsh, and Linda R. Petzold,
+"Consistent Initial Condition Calculation for Differential-Algebraic Systems",
+SIAM Journal on Scientific Computing, Vol. 19, No. 5, pp. 1495-1512, 1998.
+DOI: https://doi.org/10.1137/S1064827595289996
+
+This method modifies the algebraic variables and their derivatives to be
+consistent with the DAE constraints, while keeping the differential variables
+fixed. It uses Newton's method to solve for consistent initial values.
+
+This is the default initialization for many DAE solvers when `differential_vars`
+is provided, allowing the solver to distinguish between differential and algebraic
+variables.
+"""
+struct BrownBasicInit <: DAEInitializationAlgorithm end
+
+"""
+    struct ShampineCollocationInit <: DAEInitializationAlgorithm
+
+The Shampine collocation initialization algorithm for DAEs. This implementation
+is based on the algorithm described in:
+
+Lawrence F. Shampine, "Consistent Initial Condition for Differential-Algebraic
+Systems", SIAM Journal on Scientific Computing, Vol. 22, No. 6, pp. 2007-2026, 2001.
+DOI: https://doi.org/10.1137/S1064827599355049
+
+This method uses collocation on the first two steps to find consistent initial
+conditions. It modifies both the differential and algebraic variables to satisfy
+the DAE constraints. This is more general than BrownBasicInit but may be more
+expensive computationally.
+
+This method is useful when you need to modify all variables (both differential
+and algebraic) to achieve consistency, rather than just the algebraic ones.
+"""
+struct ShampineCollocationInit <: DAEInitializationAlgorithm end
+
+export DefaultInit, BrownBasicInit, ShampineCollocationInit