Continuous and discrete simulation
Why it matters
A model expresses relations; a simulation executes them over a horizon. A continuous tank temperature T(t) may be sampled at discrete computer times without making the physical model discrete. Inventory I changes at separate receipt/sale events. A thermostat combines continuous temperature with a discrete switch.
Learning objectives
- Tank: dT/dt=−k(T−Ta), T in °C, t in min, k in min⁻¹. Inventory: Iₙ₊₁=Iₙ+receipts−sales, integer units.
- Error: calling every computer model conceptually discrete because storage is finite. Distinguish conceptual dynamics from the numerical approximation.
Prerequisites
Courses 3.1, 4.1, 6.1 and 9.1
Concept, definitions and key terms
A model expresses relations; a simulation executes them over a horizon. A continuous tank temperature T(t) may be sampled at discrete computer times without making the physical model discrete. Inventory I changes at separate receipt/sale events. A thermostat combines continuous temperature with a discrete switch.
Notation, state, units and assumptions
Tank: dT/dt=−k(T−Ta), T in °C, t in min, k in min⁻¹. Inventory: Iₙ₊₁=Iₙ+receipts−sales, integer units.
Simulation method and procedure
A model expresses relations; a simulation executes them over a horizon. A continuous tank temperature T(t) may be sampled at discrete computer times without making the physical model discrete. Inventory I changes at separate receipt/sale events. A thermostat combines continuous temperature with a discrete switch. Tank: dT/dt=−k(T−Ta), T in °C, t in min, k in min⁻¹. Inventory: Iₙ₊₁=Iₙ+receipts−sales, integer units.
Define initial state → apply update or next event/trial → record result → verify arithmetic → compare independent evidence.
Worked simulation examples
Worked example 1
Problem, state and inputs: Tank with T(0)=30 °C, Ta=20 °C, k=.1 min⁻¹: T(10)=20+10e⁻¹≈23.68 °C. State is T; evaluate at 10 min. Verify initial value; compare with independent thermometer data to validate.
Verification: Relations versus execution.
Validation and limits: Error: calling every computer model conceptually discrete because storage is finite. Distinguish conceptual dynamics from the numerical approximation.
Worked example 2
Problem, state and inputs: Stock I₀=5 units; receipt 3 then sale 2 gives states 5,8,6. Verify 5+3−2=6; validate using a separate physical count.
Verification: No; sampling does not redefine the system.
Validation and limits: Error: calling every computer model conceptually discrete because storage is finite. Distinguish conceptual dynamics from the numerical approximation.
Common simulation error / counterexample
Error: calling every computer model conceptually discrete because storage is finite. Distinguish conceptual dynamics from the numerical approximation.
Guided practice
Classify a continuous battery-charge state sampled every hour; name state, units and horizon.
Solution / evidence
Continuous state remains conceptual; units charge and time; sampling is computational.
Trace inventory 4→+5→−3 and explain changes between events.
Solution / evidence
States 4,9,6; changes only at events.
A/B/C/D practice
Define model, simulation, state and horizon for the tank.
Solution / evidence
Equation versus its execution; state T in °C; horizon in min.
Choose state and units for a stockroom with shipments.
Solution / evidence
Integer item count; event index or clock in minutes.
Correct the claim that digital storage makes physics discrete.
Solution / evidence
Finite precision approximates a continuous model; check convergence.
Design a hybrid thermostat and independent validation test.
Solution / evidence
T in °C and switch on/off; compare cycles against measured temperature.
Mastery checks and corrective feedback
Mastery: 4/4 checks plus a correct method, units, implementation verification and one independent validation boundary. A reviewer determines mastery.
Related laboratory and next class
Time-step simulation