Time integration methods for the Heat Equation Solution u for t= Tobias Köppl TUM

Agenda: Tobias Köppl TUM2 1. Time integration methods 1.1. Implicit and explicit Onestep Methods 1.2. Convergence theory for Onestep Methods 2. The Heat Equation 2.1. Discretization of the Laplacian operator 2.2. Application of Time integration methods 2.3. Courant-Friedrichs-Levy condition 3. Outlook 4. Literature

Tobias Köppl TUM3 Notation: :is a differentiable function, which maps from the intervall into : vector space of d dimensional vectors with components, which are in : field of real numbers and whose derivative function is continous partial derivative of y with respect to t Laplacian operator of y Machine accuracy with

1. Time integration methods Tobias Köppl TUM Implicit and explicit Onestep Methods Goal: Find numerical approximations of functions which are solutions of an ODE withan initial value: (IVP)

1. Time integration methods 1.1. Implicit and explicit Onestep Methods 5Tobias Köppl TUM Example (IVP): Dahlquist‘s Testequation

Tobias Köppl TUM6 1. Time integration methods 1.1. Implicit and explicit Onestep Methods Construction of a time integration method Divide the continous intervall by discrete timepoints: Grid on for all Stepsize of grid : by a gridfunction Approximation of the solution at the gridpoints t

Tobias Köppl TUM7 1. Time integration methods 1.1. Implicit and explicit Onestep Methods Such algorithms are called Onestep Methods. Additionaly: Computation of the gridfunction by recursion, should be possible. Reason: For the computation ofonlyfrom the timepoint before is needed. Further possibility: Multistep Methods. See literature for more information f.e. [DB II] Chapter 7 or [QSS] Chapter 11

1. Time integration methods 1.1. Implicit and explicit Onestep Methods Tobias Köppl TUM8 Two popular Onestep Methods are the explicit and the implicit Euler Method: explicit Euler Method: implicit Euler Method:

1. Time integration methods 1.1. Implicit and explicit Onestep Methods Tobias Köppl TUM9 Geometry of the explicit Euler Method: The explicit Euler Method is creating a path (Euler path). t Construction of: t Compute the tangent vector t in Compute the intersection point between: and First component of this intersection point is the next value of the gridfunction Source: [DB II]

1. Time integration methods Tobias Köppl TUM Convergence theory for Onestep Methods Issue: Under which conditions does a Onestep Method converge towards the exact solution of an ODE with an initial value? Definition (local discretization error): The local discretization error of a gridfunction for a grid on the intervall is defined as: is the solution of the IVP: on

1. Time integration methods 1.2. Convergence theory for Onestep Methods Tobias Köppl TUM11 Definition: A Onestep Method is called constistent, if: for Theorem: The explicit and the implicit Euler Method are consistent. Proof: See f. e. [DB II] Chapter 4.

1. Time integration methods 1.2. Convergence theory for Onestep Methods Tobias Köppl TUM12 Defintion (convergence): A Onestep Method is called convergent towards the exact solution of an IVP, if: for Definition (global discretization error): The global discretization error is the maximum error between the computed approximations and the corresponding values of the exact solution

1. Time integration methods 1.2. Convergence theory for Onestep-Methods Tobias Köppl TUM13 Comparison of local and global Discretization error: Source: [Bun]

1. Time integration methods 1.2. Convergence theory for Onestep-Methods Tobias Köppl TUM14 Defintion (stability): A numerical algorithm is called (numerically) stable, if for all permitted input data perturbed in the size of computational accuracy ( ) acceptable results are produced under the influence of rounding and method errors. Maintheorem of Numerics: A Onestep Method is convergent, if and only if it is consistent.

1. Time integration methods 1.2. Convergence theory for Onestep-Methods Tobias Köppl TUM15 Remark: Stability of a Onestep Method is an essential condition for getting qualitatively correct solutions, when using practical stepsizes. See numerical experiment. Proof: See f. e. [Jun] Chapter 4. Theorem: The implicit Euler Method is stable for any stepsize The explicit Euler Method is only stable for small Example: If, the explicit Euler Method is a stable integrator for Dahlquist‘s testequation.

2. The Heat Equation 2.1. Discretization of the Laplacian operator Tobias Köppl TUM16 Goal: Find a numerical approximation of the two dimensional Laplacian operator in order to solve the Poisson equation: (PE)

2. The Heat Equation 2.1. Discretization of the Laplacian operator Tobias Köppl TUM17 Discretization Method: Finite Differences Main idea: Replace differential operators by difference operators Further discretization methods: Finite Volumes or Finite Elements See literature for more information f.e. [QSS] or [Wa]

2. The Heat Equation 2.1. Discretization of the Laplacian operator Tobias Köppl TUM18 Discretization of Discretizationpoints: and Uniform grid with grid parameter h

2. The Heat Equation 2.1. Discretization of the Laplacian operator Tobias Köppl TUM19 Source: [Hu]

2. The Heat Equation 2.1. Discretization of the Laplacian operator Tobias Köppl TUM20 expand in a Taylor series aroundand Discretization of the two dimensional Laplacian operator: expand in a Taylor series aroundand Discretization error: is an interior point of the unit square

2. The Heat Equation 2.1. Discretization of the Laplacian operator Tobias Köppl TUM21 In order to get a numerical solution of the (PE) we have to solve the following System of linear equations: Matrix- Vectornotation: (SLSE)

2. The Heat Equation 2.1. Discretization of the Laplacian operator Tobias Köppl TUM22 Structure of the Matrix A: A ist a sparse Matrix Use fast iterative solvers like Jacobi Method or Gauß Seidel Method for the solution of (SLSE) See f.e. [El] Source: [Hu]

2. The Heat Equation 2.1. Discretization of the Laplacian operator Tobias Köppl TUM23 Example: Solution: See numerical example

2. The Heat Equation 2.2. Application of Time integration methods Tobias Köppl TUM24 Goal: Find numerical approximations of functions which are solutions of the homogenous Heat Equation:

2. The Heat Equation 2.2. Application of Time integration methods Tobias Köppl TUM25 Solution of the Heat Equation depends on time and space. Discretization of time and space is necessary Both time and space can be discretized by an uniform grid Source: [SK] h h

2. The Heat Equation 2.2. Application of Time integration methods Tobias Köppl TUM26 Discretization of Discretizationpoints: and Uniform grid (2D) with grid parameter h Discretization of Uniform grid (1D) with stepsize k and n timepoints timepoints:

2. The Heat Equation 2.2. Application of Time integration methods Tobias Köppl TUM27 Construct a local IVP for every (IVP(ij))

2. The Heat Equation 2.2. Application of Time integration methods Tobias Köppl TUM28 Solve IVP(ij) with the implicit Euler Method: Use the discretization of the two dimensional Laplacian Operator:

2. The Heat Equation 2.2. Application of Time integration methods Tobias Köppl TUM29 We have to solve the following system of linear equations for every timestep: Use again fast iterative solvers like Jacobi Method or Gauß Seidel Method for the solution of (SLSE II) (SLSE II)

2. The Heat Equation 2.2. Application of Time integration methods Tobias Köppl TUM30 Numerical Example: (Cooling of a heated plate)

2. The Heat Equation 2.3. Courant-Friedrichs-Levy condition Tobias Köppl TUM31 (IVP(ij)) can be solved by the explicit Euler Method, too. Chapter 1.2.: explicit Euler Method is only stable for a small stepsize k Maintheorem of Numerics: Stability is an essential condition for getting qualitatively correct solutions, when using practical stepsizes.

2. The Heat Equation 2.3. Courant-Friedrichs-Levy condition Tobias Köppl TUM32 h h Theorem (CFL condition): The explicit Euler Method is a stable solver for the Heat Equation if: Proof: See [CFL] Source: [SK]

2. The Heat Equation 2.3. Courant-Friedrichs-Levy condition Tobias Köppl TUM33 Numerical Example: (Cooling of a heated plate, solved with the explicit Euler Method and a discretization which is not conform with the CFL condition)

3. Outlook Tobias Köppl TUM Let x(t) be the solution of an (IVP): Adaptive discretization algorithms, with a good error measurement are required 34 Source: [DH I] x(t) t An uniform discretization of the time axis, would lead to a Onestep Method with slow convergence

3. Outlook Tobias Köppl TUM35 Further topics: Optimization of the algorithms, solving the sparse linear equation systems, with respect to storage and number of floating point operations Traversing a certain grid efficiently (peano curves) Preconditioning of the Matrix representing the mentioned SLSE

4. Literature: Tobias Köppl TUM36 [DH I]: Numerische Mathematik I, Deuflhard/Hohmann, 2002, 3. edition [DB II]: Numerische Mathematik II, Deuflhard/Bornemann, 2002, 2. edition [Jun]: Lecture on Numerische Mathematik, Junge, 2007 [SK]: Numerische Mathematik, Schwarz/Köckler, 2006, 6. edition [QSS]: Numerische Mathematik 2, Quateroni/Sacco/Saleri, 2002, 2. edition [Wa]: Lecture on Finite Elements, Wall, 2007, 2. edition [Hu]: Numerics for computer sience students, Huckle/Schneider, 2002, 3. edition [Bun]: Lecture on numerical programming, Bungartz, 2007 [El]: Finite Elements and Fast Iterative Solvers, Elman/Silvester/Wathen, 2005, 2. ed. [Fo I]: Analysis I, Forster, 1976, 6. edition [Fo II]: Analysis II, Forster, 1976, 6. edition [Fei]: Introduction into the theory of partial differential equations, 2000 [CFL]: Über die partiellen Differentialgleichungen der mathematischen Physik, Courant, Friedrichs, Levy, 1928

Thank you for your attention!!! Tobias Köppl TUM37