Virtual Book > Mathematics and Algorithms > Advanced Numerical Differential Equation Solving in Mathematica
MATHEMATICA OVERVIEW

Advanced Numerical Differential Equation Solving in Mathematica

IntroductionIntroduction

Overview

The Design of the NDSolve FrameworkThe Design of the NDSolve Framework

Features

Common Time Stepping

Data Encapsulation

Method Hierarchy

User Extensibility

Method Classes

Automatic Selection and User Controllability

MethodMonitor

Shared Features

Some Basic Methods

ODE Integration MethodsODE Integration Methods

MethodsMethods

ExplicitRungeKutta

ImplicitRungeKutta

SymplecticPartitionedRungeKutta

Controller MethodsController Methods

Composition and SplittingComposition and Splitting

SubmethodsSubmethods
LocallyExact

DoubleStep

ExtrapolationExtrapolation

SubmethodsSubmethods
ExplicitEuler
ExplicitMidpoint
ExplicitModifiedMidpoint
LinearlyImplicitEuler
LinearlyImplicitMidpoint
LinearlyImplicitModifiedMidpoint

FixedStep

OrthogonalProjection

Projection

StiffnessSwitching

ExtensionsExtensions

Method Plug-in Framework

Partial Differential EquationsPartial Differential Equations

The Numerical Method of LinesThe Numerical Method of Lines

Introduction

Spatial Derivative Approximations

Boundary Conditions

Spatial Error Estimates

Boundary Value ProblemsBoundary Value Problems

Shooting Method

Chasing Method

Boundary Value Problems with Parameters

Differential-Algebraic EquationsDifferential-Algebraic Equations

Introduction

Index of a DAE

Treatment of Differential Equations

DAE Solution Methods

Index Reduction for DAEsIndex Reduction for DAEs

Index Reduction AlgorithmsIndex Reduction Algorithms

Pantelides Method
Structural Matrix Method

Constraint MethodsConstraint Methods

Dummy Derivatives
Projection

Consistent InitializationConsistent Initialization

Collocation MethodCollocation Method

Collocation Direction
Collocation Points
Collocation Range

QR Method

Block Lower Triangularization (BLT) Method

Reinitialization with Events

Examples

MethodsMethods

IDA

StateSpaceStateSpace

Block Lower Triangular (BLT) Order

Delay Differential EquationsDelay Differential Equations

Introduction

Comparison and Contrast with ODEs

Propagation and Smoothing of Discontinuities

Storing History Data

Method of Steps

Examples

Error ControlError Control

ScaledVectorNorm

Stiffness DetectionStiffness Detection

Overview

Introduction

Linear Stability

"StiffnessTest" Method Option

"NonstiffTest" Method Option

Examples

Option Summary

Structured SystemsStructured Systems

LotkaVolterraLotkaVolterra

Introduction

Explicit Euler

Backward Euler

Projection

Splitting

Symplectic Euler

Flows

RigidBodyRigidBody

Introduction

Manifold Generation and Utility Functions

Method ComparisonMethod Comparison

Adams Multistep Method
Euler and Implicit Midpoint Methods
Orthogonal Projection Method
Projection Method
Projecting One Constraint
Projecting Multiple Constraints
"Splitting" Method
Solution

Events and Discontinuous SystemsEvents and Discontinuous Systems

Event Detection and LocationEvent Detection and Location

Event Detection Methods

Event Location Methods

Event ActionsEvent Actions

Changing State Variables

Events and Discontinuous Differential EquationsEvents and Discontinuous Differential Equations

Sliding Mode

Discontinuity Signature

Examples

Components and Data StructuresComponents and Data Structures

Introduction

Example

Creating NDSolve`StateData ObjectsCreating NDSolve`StateData Objects

ProcessEquations

Reinitialize

Iterating Solutions

Getting Solution Functions

NDSolve`StateData PropertiesNDSolve`StateData Properties

Solution Data

NumericalFunction

DifferentialEquations Utility PackagesDifferentialEquations Utility Packages

InterpolatingFunctionAnatomy

NDSolveUtilities

References

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