Technology
TDSE™ addresses one of the most stubborn inefficiencies in time-domain simulation: the repeated solution of linear subsystem dynamics that have not actually changed. When a subsystem remains linear over the simulation horizon, its contribution can be characterized once as response data and then reused inside the host integration loop.
What TDSE Does
The central claim is simple but powerful. In many workflows, the same linear subsystem is solved again at every time step even though its dynamics do not change. TDSE™ replaces that repeated solve path with a reusable operator built from subsystem response data and evaluated inside the host solver.
When a subsystem is linear and time-invariant over the simulation horizon, its response can be computed once and reused rather than re-solved inside every step of the full system simulation. That is where TDSE™ creates its first major leverage point.
TDSE™ is not a standalone simulator. It acts as an operator layer that plugs into existing ODE, DAE, and semi-discrete workflows through solver-compatible equations. Its value comes from changing the runtime model without discarding the host stack.
How It Works
The structure is straightforward, but the consequence is far-reaching: isolate the reusable subsystem, characterize its response once, and then let the resulting equivalent equation advance inside the host integrator. The decisive step is turning subsystem response into a causal operator that can live inside the surrounding time-stepping scheme.
Identify the part of the model whose dynamics can be reused while leaving the remaining system nonlinear, time-varying, controlled, or otherwise tightly coupled.
Build a causal time-domain operator from independent response and impulse response data for the selected subsystem interfaces.
Insert the reusable contribution into the host solver as an equivalent equation evaluated during time stepping.
Equation Form
The resulting operator is evaluated through an equivalent equation of the form, combining independent response, history response, and direct terms into a single solver-compatible expression that advances inside the host time-stepping scheme.
Reusable baseline response carried in from precomputed subsystem behavior.
Convolution term that brings retained interface history into the current evaluation.
Instantaneous interface contribution aligned with the host solver's current step.
The New Scaling Law
Once reusable subsystem dynamics move into operator form, the dominant cost of time-domain simulation shifts away from repeated global solves and toward interface dimension, response history, and host-solver coupling. That is not just an implementation detail. It is a different scaling regime.
In simplified form, the dominant scaling is
Np is the number of exchanged interface or port variables. In dense coupling, operator work grows with interface dimension rather than with the full size of the reusable subsystem.
Nh is the retained response-history length. Truncation and stored history directly set the size of the convolution workload and make runtime cost easier to tune and reason about.
The operator form is structurally more favorable for parallel execution than repeated monolithic subsystem solves. Interface-wise convolution, retained-history evaluation, and multi-operator application can often be decomposed across cores or accelerators more naturally. Realized gains still depend on host-solver scheduling, memory bandwidth, and interface organization, but the computational structure becomes far more parallel-ready.
What Makes It Practical
TDSE™ is meant to survive contact with real simulation workflows. That means the equation form, the response source, the reconstruction path, the retained history, and the numerical conditioning all have to hold up inside a real solver loop, not just in a derivation.
Equation Form
The reusable contribution is assembled from independent-response, history-response, and direct-response terms so it can advance inside standard host integration schemes. That matters because the operator has to appear in a form the surrounding solver can actually evaluate, not merely in a form that looks elegant on paper.
Response Sources
Response data can come from transient simulation, inverse frequency transformation, or modal decomposition. TDSE™ is designed to reuse trusted analysis outputs in runtime form rather than replace the pipeline that produced them. That keeps the operator layer compatible with established characterization workflows instead of demanding a new ecosystem.
Causality
Finite frequency sampling and reconstruction artifacts can produce non-causal impulse responses. Practical operator runtime depends on restoring physically meaningful causal behavior before deployment. Without that correction, the operator may look persuasive offline and still behave incorrectly once inserted into live time-domain integration.
Conditioning
Truncation, interpolation, resampling, and variable-step alignment determine whether an operator remains numerically stable once it is embedded in a larger solver loop. Practical deployment depends as much on these conditioning choices as on the abstract operator derivation itself.
History
The retained response history is not just a complexity term. It determines memory footprint, convolution workload, and the balance between accuracy and runtime cost. Choosing that history length is therefore part of the implementation, not an afterthought.
Variable Step
Real deployment involves variable-step host solvers. The operator must handle step changes gracefully, either through resampling, interpolation, or step-aware reconstruction. That is not an optional feature—it is essential for practical use in production simulation environments.