Extending explicit and linearly implicit ODE solvers for index-1 DAEs

• Single-step solution for DAE initialization and solving.
• Combines perturbation initialization approach and hyperbolic tan switching function.
• Solves stiff index-1 DAE with explicit and linearly implicit solvers.
• Loosens restriction on initial guesses for algebraic variables.
• Solves discretized PDEs and implicit ODEs.

Nonlinear differential-algebraic equations (DAE) are typically solved using implicit stiff solvers based on backward difference formula or RADAU formula, requiring a Newton–Raphson approach for the nonlinear equations or using Rosenbrock methods specifically designed for DAEs. Consistent initial conditions are essential for determining numeric solutions for systems of DAEs. Very few systems of DAEs can be solved using explicit ODE solvers. This paper applies a single-step approach to system initialization and simulation allowing for systems of DAEs to be solved using explicit (and linearly implicit) ODE solvers without a priori knowledge of the exact initial conditions for the algebraic variables. Along with using a combined process for initialization and simulation, many physical systems represented through large systems of DAEs can be solved in a more robust and efficient manner without the need for nonlinear solvers. The proposed approach extends the usability of explicit and linearly implicit ODE solvers and removes the requirement of Newton–Raphson type iteration.