Numerical Integration methods are used to solve initial value problems of differential equations. For example, integrating an accelerometer output twice can provide the position of an object over time. Integration methods make different assumptions about how the signal changes between the discrete points. The higher the order of the integration method, the more accurate the results are. Different integration methods may use different numbers of function evaluations to determine the value at the next time step. These methods are used to step from a value at one timestep to a value at a second timestep. Using numerical integration has the potential to introduce instability and errors, but it is usually the only way to solve very complicated systems of equations. It is also very useful for systems with nonlinear input. For fast systems with a very high natural frequency, you need to be careful about selecting an integration method. For some applications, combining different integration methods leads to a better function evaluation. For Example, you could use Euler’s method for the acceleration-to-velocity integration and then the Tustin method for the velocity-to-position integration. Numerical integration methods are used to solve state-space equations. The more information about the modeled system that you can bake into the integration method the better your results will be.[^7]
[[Euler’s Method]]
[[Runge-Kutta Methods]]
[[Heun’s Predictor-Corrector]]
[[Modified Euler Mid-Point]]
[[Comparing Numerical Integration Methods]]
[[Second-Order Trapezoidal Integration]]
[[Tustin Discretization method]]
[[Forward Euler]] – quick dirty integration scheme
[[Backward Euler]] – quick dirty integration scheme
[[Adaptive Integration Methods]]
[[Implicit Integration Methods]]
[[Stability Region of Numerical Integration Methods]]
[[Adams-Bashforth Methods]]
[[Trapezoidal Integration]]
[[Euler’s Method with Tustin Discretization]]
[[2nd Order Adams-Bashforth with Rectangular]]
[[2nd Order Adams-Bashforth with Trapezoidal]]
[[Rectangular Integration]]
[[Initial Value Problems]]
[[Differential Equation Solver Stability]] – the integration methods solve differential eq’s but they must be numetrically stable.
State-Space Simulation]] – solved with a numerical integration method.
[[Analytical Methods]] – opposite to numerical methods
[[Numerical Differentiation Methods]] -opposite
State-Space Model]] – state-space models are numerically integrated to obtain the time-series dynamics.
[[Discrete Integration]] – used for discrete timesteps
Point Reactor Kinetics Model – numerical integration methods that can’t accurately model point kinetics cannot be used for space-time kinetic reactor models.
RELAP5– uses a semi-implicit solver for solving the ODEs associated with the nuclear reactor.
[[BDF2 Method]]
[[Crank-Nicholson Method]]
[[Fixed-Step Integration Methods]]
[[SPICE]] – tracks behaviors of capacitances and inductances using numerical integration
[[Circuit Simulators]] – typically use integration methods for simulations.
[[Teetering Rotors]] – the high-frequency dynamics may be filtered by the integration methods.
[[Flight Path Angle Controller]] – uses discrete integration for the pitch-rate feedback.
[[Shooting Algorithm]] – uses numerical integration in each iteration of the algorithm
[[Numerical Integration Difference Equations]] – examples of difference equations for filter implementation
[[Stiffness Ratio]] – stiff systems are more difficult to solve
[[COSMOS]] – uses numerical integration in first part of metabolite concentration modeling
[[Auto-GCAS]] – root of auto-GCAS systems is the integration of aircraft model
[[Errors in Integration Methods]]
[[Explicit Integration Methods]]
[[Orbital Dynamics]] – numerical integration methods are used in orbital dynamics
[[Volume-Conserving Numerical Integration]] – used in phase space
[[Symplectic Integration]]
[[CFL Condition]] – must be used for fluid flow to ensure that the step size is small enough for the discretized problem volume
[[Deriving the Verlet Integration]] – verlet integration is used in particle simluations
- peterzipfelDoFSimulations2022
- RungeKuttaIntegratorOverview
- NumericalApproachStudying2018
- witkinPhysicallyBasedModeling
- mcfarlandStabilityDiscreteIntegration
- IntroductionStateSpaceEquations
- verizonMasterComplexitySpaceflight2024
Backlinks:
[[Designing code for discrete systems]]
[[Discrete Time Functions]]
[[Orbital Dynamics Simulations]]
State-Space Model