Glossary

Access time

(ISO) The time interval between the instant at which a call for data is initiated and the instant at which the delivery of the data is completed.

Adams–Moulton

The Adams–Bashforth methods are explicit methods. The Adams–Moulton methods are similar to the Adams–Bashforth methods except that they are implicit methods. An s-step Adams–Moulton method can reach order s + 1, while an s-step Adams–Bashforth methods has only order's.

BDF

The Backward Differentiation Formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations. They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed times, thereby increasing the accuracy of the approximation. These methods are especially used for the solution of stiff differential equations.

Computation

Computation is a general term for any type of process, algorithm or measurement; this often includes but is not limited to digital data. This includes phenomena ranging from human thinking to calculations with a more narrow meaning. Computation is a process following a well-defined model that is understood and can be expressed in an algorithm, protocol, network topology, etc. Computation is also a major subject matter of computer science: it investigates what can or cannot be done in a computational manner.

Convergence

The tendency for a solution to approach the exact solution within well-defined and specified tolerances, for example, by reducing the time step.

CPU performance

Refers to user CPU time (this is our primary focus).

CPU time

As reported in the solver output, CPU time generally refers to the time that a processor spends on the user's application; it excludes system and I/O wait time and other idle time. For parallel systems, CPU time means different things on different systems. Some systems report CPU time summed across all threads, while others do not. It is best to use "elapsed" or "wall" time for parallel applications. It doesn't count I/O or time spent running other programs and it can be broken up into system time, and user time.

DASSL

DASSL is a Differential Algebraic System Solver code (Linda R. Petzold, 1982). This code implements the Backward Differentiation Formulas of orders one through five to y and y'. The derivatives are approximated by backward differentiation formulae (BDFs), and the resulting nonlinear system at each time-step is solved by Newton's method. This code is good for stiff ODEs and for DAEs of moderate size, where it is appropriate to treat the Jacobian matrix with dense or banded direct LU decomposition. For large-scale stiff ODE and DAE problems, the user should consider DASPK. For ODE or DAE problems which must stop at the root of a given function of the solution, the user should consider DASKR.

Differential-Algebraic Equation (DAE)

A set of equations that includes both differential and algebraic equations. A DAE is classified in terms of its index, a positive integer, which is related to the minimum number of differentiations needed to transform a DAE to an ODE form.

Elapsed time (response time or wall-clock time)

The time that passes from the moment when a program starts to run to the moment when it finishes. It counts everything (disk and memory accesses, I/O, etc.). It’s a useful number, but often not good for comparison purposes.

Error

Deviations from the correct solution, primarily due to: poor modeling; discretization (such as insufficiently fine mesh, poor elements, or insufficiently short time steps); and round off and truncation (depending on numerical representation, ill-conditioning, or the solution algorithms).

Error Estimate

An estimation of the error in the numeric solution to a problem, either locally or globally, primarily for use by an adaptive mesh refinement.

Euler

Euler method, named after Leonhard Euler (1707-1783), is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic kind of explicit method for numerical integration of ordinary differential equations. Euler's method is a one-step method. A simple multistep method is the two-step Adams–Bashforth method.

Execution time (or CPU execution time)

It’s the actual time a CPU spends computing for a specific task. It’s the sum of the system CPU time (time spent in OS on behalf of your program) and of the user CPU time (time spent in executing lines of code inside your program). It does not count I/O or time spent running other programs. See User time (or user CPU time).

Explicit and Implicit Methods

Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one.

Explicit Euler

Explicit Euler is also called Forward Euler.

Function

A function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can also be elements from any given set.

Ill-Conditioning

An ill-conditioned system is sensitive to small changes in the inputs and is susceptible to round off errors.

Implicit Euler

Implicit Euler is also called Backward Euler. Implicit Euler could be stable even when physics is not. Implicit Euler damps out motion unrealistically.

Index (for DAE)

See Differential-Algebraic Equation (DAE).

Initial Condition

The starting values for the dependent variables in a time-dependent analysis and for nonlinear iterations or other iterative solvers.

Initial Value Problem

In the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution. In physics or other sciences, modeling a system frequently amounts to solving an initial value problem. In this context, the differential equation is an evolution equation specifying how, given initial conditions, the system will evolve with time.

Iterative Solver

A solver for a system of linear equations that uses an iterative method, computing a sequence of more and more accurate approximations to the solution. Each step in this sequence is one linear iteration. This should not be confused with the Newton iterations (nonlinear iterations) that occur in the solution of a nonlinear system of equations. Compare to Direct Solver and Nonlinear Iteration.

Jacobian matrix

Jacobian matrix is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. It is a matrix containing the first derivative of a vector-valued function of a vector variable. In particular, it is the derivative of the residual vector with respect to the solution vector. When used in this narrower sense, the term stiffness matrix is sometimes used.

Latency (ISO)

The time interval between the instant at which a CPU's instruction control unit initiates a call for data and the instant at which the actual transfer of the data starts. Synonym: waiting time.

Linear Multistep Methods

Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge-Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of linear multistep methods, a linear combination of the previous points and derivative values is used.

Linear PDE

An equation where both sides are sums of a known function, the unknown functions, and their partial derivatives, multiplied by known coefficients that only depend on the independent variables. Other PDEs are called nonlinear.

Linear System

A linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, …

LSODA

Livermore Solver for Ordinary Differential Equations with Automatic Method Switching Solver (Petzold, 1983). For Ordinary Differential Equations (ODE), Switching Automatically Between Stiff and Non-stiff Methods. Solving initial value problems for stiff or non-stiff systems of first-order ordinary differential equations (ODEs). LSODA differs from the other integrators (except LSODAR) in that it switches automatically between stiff and nonstiff methods. This means that the user does not have to determine whether the problem is stiff or not, and the solver will automatically choose the appropriate method. It always starts with the nonstiff method. LSODA automatically selects a stiff or nonstiff method. It may switch between the two methods during the simulation, in case the stiffness of the system changes. This is the default method used in ode and especially well-suited for simple problems.

Newton’s Method

An iterative solver method, also called the Newton-Raphson method, for solving nonlinear equations.

Nonlinear System

Nonlinear system is a system which is not linear, that is, a system which does not satisfy the superposition principle, or whose output is not directly proportional to its input. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. A non-homogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system of multiple variables.

Numerical Stability

Numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm. A related phenomenon is instability. Typically, algorithms would approach the right solution in the limit, if there were no round-off or truncation errors, but depending on the specific computational method, errors can be magnified, instead of damped, leading to large errors.

ODE

Ordinary Differential Equation. An equation involving functions and their derivatives. The derivatives are with respect to one independent variable only. ODE is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable. Compare to Partial Differential Equation (PDE)

PDE

Partial Differential Equation. An equation involving functions and their partial derivatives. PDE is an equation that includes derivatives with respect to more than one independent variable. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic. Just as ordinary differential equations often model dynamical systems, partial differential equations often model multidimensional systems. Compare to Ordinary Differential Equation (ODE).

Response time

See Elapsed time (response time or wall-clock time).

RK

Runge-Kutta 1st Order integration. Explicit One-Step Solvers for Ordinary Differential Equations (ODE). Solving initial value problems for non-stiff systems of first-order ordinary differential equations (ODEs).

RK4

Runge-Kutta 4th Order integration. RK4 Solve System of ODE (Ordinary Differential Equation)s by Euler’s Method or Classical Runge-Kutta 4th Order Integration. Solving initial value problems for systems of first-order ordinary differential equations (ODEs) using Euler’s method or the classical Runge-Kutta 4th order integration.

Runge–Kutta Methods

The Runge–Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians Carl Runge (1856-1927) and Martin W. Kutta (1867-1944).

Solver

A solver is a generic term indicating a piece of mathematical software, possibly in the form of a stand-alone computer program or as a software library that 'solves' a mathematical problem. A solver takes problem descriptions in some sort of generic form and calculates their solution.

Stiff Equation

A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has been proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution.

Stiff System

We call a linear system stiff if it is stable and its eigenvalues vary a lot in terms of their real parts. Non-linear systems are called stiff if they are stable and exhibit both fast and slow modes in their behavior. The linearization of such systems leads to stiff linear systems.

System performance

Refers to elapsed time for an unloaded system.

System time (or system CPU time)

Time spent by the operating system on behalf of a user’s program.

Timing (IEEE)

The process of estimating or measuring the amount of execution time required for a software system or component. Contrast with sizing.

Timing analyzer (IEEE)

A software tool that estimates or measures the execution time of a computer program or portion of a computer program, either by summing the execution times of the instructions along specified paths or by inserting probes at specified points in the program and measuring the execution time between probes.

User time (or user CPU time)

Time spent by the processor executing a user’s program. Time spent executing the lines of code that are "in" our program.

Wall clock time

Total elapsed time it takes to complete an application.