ShaneLogic/dae-solver-configuration-matlab

DAE Solver Configuration in MATLAB

This skill configures MATLAB's ode15s solver for time evolution simulations of discretized semiconductor device models.

When to Use

• You have a discretized system of DAE equations
• You need to perform time evolution simulation
• You're working in the MATLAB environment
• You need adaptive step size and order control
• Error tolerances must be specified for accuracy control

Prerequisites

• Discretized system of equations available
• State variables identified (e.g., P, φ, n, p)

Solver Configuration Procedure

1. Solver Selection

• Tool: MATLAB's ode15s
• Method: Based on Numerical Differentiation Formulae (NDF) of orders 1-5, which are related to Backward Differentiation Formulae (BDF)
• Key Features:
  • Variable step size to meet error tolerances
  • Variable order (1-5) for efficiency
  • Suitable for stiff DAE systems

2. System Assembly

Goal: Minimize computational cost by eliminating superfluous variables.

Eliminate the following variables from the state vector:

• FP (flux-related variable)
• jn (electron current density)
• jp (hole current density)
• E (electric field)

3. State Vector Construction

The state vector u(t) is a column vector of length 4N + 4:

u(t) = [ P(t)^T, φ(t)^T, n(t)^T, p(t)^T ]^T

Where:

• P(t): Column vector of anion vacancy concentrations at N+1 grid points
• φ(t): Column vector of electric potential at N+1 grid points
• n(t): Column vector of electron densities at N+1 grid points
• p(t): Column vector of hole densities at N+1 grid points

4. Problem Formulation

Write the system in the standard DAE form:

M * du/dt = f(t, u)

or:

F(t, u, du/dt) = 0

Ensure the form matches what ode15s expects for your specific problem type.

Required Parameters

• Relative tolerance: Specify for accuracy control (e.g., RelTol)
• Absolute tolerance: Specify per variable or as a scalar (e.g., AbsTol)
• Time span: [t0, tf] for the simulation

Output

A configured MATLAB solver setup ready for time integration via ode15s.