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1 month ago Arbitrary ODE Solver Test Cases
This document provides test cases for a new ODE solver using the Van der Pol oscillator and the Rossler attractor. Each test case includes the problem definition, parameters, initial conditions, and expected behavior.
In each case, the only provided inputs are a string value of the ode system, initial conditions, known constant values, and the time span for the initial value problem. No c++ functions have been compiled a priori as these numerical functions (and, if needed, their Jacobians) should be constructed by the arbitrary solver.
Explanation
-
Van der Pol Oscillator:
- The system is defined by converting the second-order ODE into two first-order ODEs.
- Parameters and initial conditions are specified.
- The expected behavior is a limit cycle.
-
Rossler Attractor:
- The system is defined by three first-order ODEs.
- Parameters and initial conditions are specified.
- The expected behavior is chaotic with a fractal structure.
These test cases are designed to assess initial functionality and performance of the new ODE solver on both periodic and chaotic systems.
Test Case 1: Van der Pol Oscillator
Problem Definition
The Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It is described by the following second-order differential equation:
$$ \frac{d^2x}{dt^2} - \mu (1 - x^2) \frac{dx}{dt} + x = 0 $$
This can be converted into a system of first-order ODEs:
$$ \begin{cases} \frac{dy_0}{dt} = y_1 \ \frac{dy1}{dt} = \mu (1 - y{0}^2) y_1 - y_0 \end{cases} $$
Parameters
- $(\mu = 1.0)$ (standard non-linearity parameter)
Initial Conditions
- $(y_0(0) = 2.0)$
- $(y_1(0) = 0.0)$
Time Span
- $(t \in [0, 20])$
Expected Behavior
The Van der Pol oscillator should exhibit a limit cycle behavior for $(\mu = 1.0)$. The solution should converge to a periodic orbit.
Input Parameters for Solver
- ODE System String: "dy0_dt = y1; dy1_dt = mu(1-y0y0)*y1-y0"
- Constant Values: {{"mu", 1.0}}
- Initial Conditions: {2.0,0.0}
- Time Span: [0, 20]
Obtained Solution
Combined Results
$y_0$ Results
$y_1$ Results
Test Case 2: Rossler Attractor
The Rossler Attractor is a system of three coupled non-linear differential equations. It is described by:
$$ \begin{cases} \frac{dy_0}{dt} = -y_1 - y_2 \ \frac{dy_1}{dt} = y_0 + ay_1 \ \frac{dy_2}{dt} = b + y_2(y_0 - c) \end{cases} $$
Where:
- $( y_0(t) ), ( y_1(t) ), ( y_2(t) )$ are the state variables.
- $( a ), ( b ), ( c )$ are parameters that define the behavior of the system.
Parameters
- $(a = 0.2)$
- $(b = 0.2)$
- $(c = 5.7)$
Initial Conditions
- $(y_0(0) = 0.0)$
- $(y_1(0) = 1.0)$
- $(y_2(0) = 1.05)$
Time Span
- $(t \in [0, 100])$
Expected Behavior
The Rossler Attractor should exhibit chaotic behavior with a fractal structure in the state space. The trajectory should be sensitive to initial conditions and parameters.
Input Parameters for Solver
- ODE System string: "dy0_dt = -y1-y2;\ndy1_dt = y0+ay1;\ndy2_dt = b+y2(y0-c)"
- Constant Values: {{"a", 0.2}, {"b",0.2}, {"c", 5.7}}
- Initial Conditions: {0.0, 1.0, 1.05}
- Time Span: [0,100]
Obtained Solution
Combined Results
$y_0$ Results
$y_1$ Results
$y_2$ Results
Summary & Importance
Initial testing demonstrates that strings representing ODE systems can be effectively implemented and yield solutions to some classic problems. Both the solver and the ODE constructor leverage base classes already defined in svFSIplus. This added functionality may help to extend the solver package to be more modular and avoid hard-coding of ODE physics. This added functionality requires one additional third party package (exprtk.h) and, optionally, a symbolic engine library (symengine) which is used in a preprocessing stage. This work is crucial for the subsequent implementation of a systems biology metabolic model.
Next Steps
The obtained solutions for these two test cases will be validated against benchmark ODE solver packages to test the accuracy of solutions. If satisfactory, efficiency testing and code clean-up will be started to speed up numeric evaluations (if needed) and provide performance metrics for solutions.
ktbolt
Use Case
Often for perfusion and metabolic modeling, different systems of differential algebraic equations are included as reaction terms to represent kinetic components of overall metabolism. Often hypotheses about metabolic functionality require the form of these equations to be changed substantially (e.g. the inclusion of fatty acid oxidation in a cardiomyocyte model versus glucose oxidation alone). Therefore, users interested in simulating multiple models for tissue perfusion may require a flexible framework to easily change governing reaction equations.
Problem
Currently, any changes to governing equations require recompiling svFSIplus to integrate. Futhermore, the lack of modularity hinders incorporation of complex reaction terms in more complex advection-diffusion-reaction systems.
Solution
We shall introduce flexible string parser to allow for inclusion of arbitrary algebraic and ordinary differential equations into on-demand c++ functions. Such an approach is attractive for general purpose uses and computational efficiency (as opposed to symbolic solvers which may be large third party packages or too slow for integration in a governing PDE equation).
Alternatives considered
Symbolic approaches may be considered if direct construction of c++ functions cannot easily be accomplished (although this is not desirable).
Additional context
Specifically this will be important for perfusion and metabolic modeling although there are many more general use cases.
Code of Conduct