Efficient Simulation of Physical System Models Using Inlined Implicit Runge-Kutta Algorithms

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Efficient Simulation of Physical System Models Using Inlined Implicit Runge-Kutta Algorithms. Vicha Treeaporn Department of Electrical & Computer Engineering The University of Arizona Tucson, Arizona 85721 U.S.A. Topics. Introduction Techniques for Simulation Results An Application.

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### Efficient Simulation of Physical System Models Using Inlined Implicit Runge-Kutta Algorithms

Vicha Treeaporn

Department of Electrical & Computer Engineering

The University of Arizona

Tucson, Arizona 85721 U.S.A

Topics
• Introduction
• Techniques for Simulation
• Results
• An Application
Introduction
• Stiffness
• Widely varying eigenvalues
• Explicit algorithms
• Straightforward to implement
• Step size limited by numerical stability
• Implicit algorithms
• More difficult to implement
• Needed to simulate stiff systems
• May use larger step sizes
Inline-Integration
• Merges the integration algorithm with the model
• Eliminates differential equations
• Results in difference equations (∆Es)
• Easily implement implicit algorithms

Evaluate at

Eliminate

derivatives

Integrator equations

Sorting
• 10 equations immediately causalized
• Need to perform tearing
• Make assumptions about variables being ‘known’
Tearing

Tearing

variable

Residual Eq.

Tearing

Residual Eq. #2

Tearing variable #2

Tearing
• Completely causalized equations
• 2 iteration variables, vc and i1
• Could use this set of equations for simulation
• Want step-size control
Step-Size Control
• Want larger step sizes
• Reduce the overall computational cost
• Maintain desired accuracy
• Compute error estimate
• Embedding method
• Shares computations with original method
Step-Size Control
• Explicit RKs
• Embedding methods have been found
• Implicit RKs
• Difficult problem
• Algorithms are compact
• Can find embedding methods using two steps
• Linear polynomial approximation
HW-SDIRK Embedding
• 3rd-order accurate
• Behaves like an explicit method
• May unnecessarily restrict step size for stiff systems
• Search for an alternate embedding method
Alt. HW-SDIRK Embedding
• 3rd-order accurate
• Implicit method
Alt. HW-SDIRK Embedding

Stability Domain

Damping Plots

Lobatto IIIC(6)
• No embedding method exists
• Expensive to perform step size control
• Can search for an embedding method
Lobatto IIIC(6) Embedding Method
• 5th-order accurate
• A-Stable
• Large asymptotic region
Lobatto IIIC(6) Embedding Method

Stability Domain

Damping Plots

### Numerical Experiments

Numerical Experiments
• Tested various algorithms with selected benchmark ODEs
• Implemented in Dymola/Modelica
ODE Set B

Inlined with HWSDIRK and

alternate error method

ode15s

ODE Set B

Error estimate stays near 10-3

Step size grows and

shrinks appropriately

ODE Set D

Inlined with Lobatto IIIC(6)

ode15s

### An Application

An Application
• Real-Time, Limited Resources
• Embedded control systems
• Model Predictive