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Advanced Numerical Differential Equation Solving in the Wolfram Language

Advanced Numerical Differential Equation Solving in the Wolfram Language

Introduction

Overview

The 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 Methods

Methods

ExplicitRungeKutta

ImplicitRungeKutta

SymplecticPartitionedRungeKutta

Controller Methods

Composition and Splitting

Submethods

LocallyExact

DoubleStep

Extrapolation

Submethods

ExplicitEuler

ExplicitMidpoint

ExplicitModifiedMidpoint

LinearlyImplicitEuler

LinearlyImplicitMidpoint

LinearlyImplicitModifiedMidpoint

FixedStep

OrthogonalProjection

Projection

StiffnessSwitching

Extensions

Method Plug-in Framework

Partial Differential Equations

Finite Element Method Overview

Solving Partial Differential Equations with Finite Elements

Element Mesh Generation

Element Mesh Visualization

Finite Element Programming

The Numerical Method of Lines

Introduction

Spatial Derivative Approximations

Boundary Conditions

Spatial Error Estimates

Boundary Value Problems

Shooting Method

Chasing Method

Boundary Value Problems with Parameters

Differential-Algebraic Equations

Introduction

Index of a DAE

Treatment of Differential Equations

DAE Solution Methods

Index Reduction for DAEs

Index Reduction Algorithms

Pantelides Method

Structural Matrix Method

Constraint Methods

Dummy Derivatives

Projection

Consistent Initialization

Collocation Method

Collocation Direction

Collocation Points

Collocation Range

QR Method

Block Lower Triangularization (BLT) Method

Reinitialization with Events

Examples

Methods

IDA

StateSpace

Block Lower Triangular (BLT) Order

Delay Differential Equations

Introduction

Comparison and Contrast with ODEs

Propagation and Smoothing of Discontinuities

Storing History Data

Method of Steps

Examples

Error Control

ScaledVectorNorm

Stiffness Detection

Overview

Introduction

Linear Stability

"StiffnessTest" Method Option

"NonstiffTest" Method Option

Examples

Option Summary

Structured Systems

LotkaVolterra

Introduction

Explicit Euler

Backward Euler

Projection

Splitting

Symplectic Euler

Flows

RigidBody

Introduction

Manifold Generation and Utility Functions

Method 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 Systems

Event Detection and Location

Event Detection Methods

Event Location Methods

Event Actions

Changing State Variables

Events and Discontinuous Differential Equations

Sliding Mode

Discontinuity Signature

Examples

Components and Data Structures

Introduction

Example

Creating NDSolve`StateData Objects

ProcessEquations

Reinitialize

Iterating Solutions

Getting Solution Functions

NDSolve`StateData Properties

Solution Data

NumericalFunction

DifferentialEquations Utility Packages

InterpolatingFunctionAnatomy

NDSolveUtilities

References

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