1. 7. Introduction to the numerical integration of PDE. As an example, we consider the following PDE with one variable; Finite difference method is one of numerical method for the PDE.

2. Accuracy requirements Usually  is more restricted by stability than by accuracy. Notation for the discretization of

3. Summary of the key concept on numerical method for PDE. Global discretization error Local truncation error  : The amount that the exact solution of PDE fails to satisfy the the finite difference equation. ex.) One-step method. Definitions: (Consistency) Definitions: (Convergence)

4. Definitions: (Zero-Stability - a stability criteria with h ! 0 ) Definitions: (Absolute Stability - a stability criteria with a fixed h) Remark: Purpose of stability analysis is to determine  0 (h) which guarantee that the perturbation does not glow. This is the case if Note that for the case of Zero-stability, the dimension (size) of the matrix C 0 (h) increase as n ! 1.

5. Theorem: (Lax’s equivalence theorem) Convergence ) Zero (or Absolute) - stability. Zero (or Absolute) - stability and Consistency ) Convergence. l p norm and l 1 norm are defined by

6. Some explicit integration method for a linear wave equation. Some basic schemes for are presented. (1) FTCS (Forward in Time and Central difference Scheme). (2) Lax (- Friedrich) scheme. (3) Leap-Flog scheme (4) Lax-Wendroff scheme (5) 1 st order upwind scheme

7. Some explicit integration method for a linear wave equation continued. Lax, Lax-Wendroff, 1 st order upwind schemes can be understood as FTCS scheme +.Diffusion term. (1) Explicit Euler scheme (FTCS : Forward in Time and Central difference). (2) Lax (- Friedrich) scheme. (3) Lax-Wendroff scheme. (4) 1 st order upwind scheme (Can be used also for negative c.) (weakest diffusion)

8. von Neumann stability analysis. A method to analyze the stability of numerical scheme for linear PDE (assuming equally spaced grid points and periodic boundary condition). Consider that the finite difference equation has the following solution. Then the perturbation can be also written Substituting the Fourier transform above, we have Amplification factor g(  ) is defined by

9. The von Neumann condition for zero stability: The von Neumann condition for absolute stability:

10. Another derivation of von Neumann stability condition. Then we apply absolute stability condition for the ODE. l 2 norm. Parceval’s theorem. Test problem: Definition: The region of absolute stability for a one-step method is the set Characteristic polynomial Q: Therefore for the PDE, the region of absolute stability is the set Note Recall:

11. Characteristic polynomials for a particular FD scheme. A solution to the polynomial becomes an amplification factor for each mode  If the exact values are known, conveniently shows amplification rate and phase error of each mode. A substitution of to a FD equation, and the mode decomposition results in a equation for each Fourier component. Then, substituting we have characteristic polynomials. Note:  = 0, corresponds to low frequency (long wavelength).  = , corresponds to high frequency (short wavelength).

12. Define a flux at the interface j–1/2, j+1/2,, then discretize PDE using explicit Euler scheme in time as Finite volume discretization Another concept for deriving finite difference approximation suitable for the conservation law; PDEs of the conservative form j+1jj–1 j–1/2j+1/2 unit volume However, there is no grid point (no data!) at j-1/2, j+1/2…  Use to evaluate One can rewrite explicit Euler, Lax, Lax-Wendroff, 1 st order upwind using (1) Explicit Euler :

13. (2) Lax : (3) Lax-Wendroff : (4) 1 st order upwind : (1) Explicit Euler :

14.  – scheme : A parametrization of representative linear schemes. (Van Leer) For a linear PDE, write a Taylor expansion in time Approximate the second term in RHS as And the third term as j+1jj–1 j–1/2j+1/2 j–2

15. Deriving a FD scheme explicitly for w n j, one finds that the coefficients of w n j Are effectively proportional to  – 2  | |. Hence one parameter may be eliminated. A choice results in the form of  – scheme derived by Van Leer.  – scheme becomes Different from the Van Leer’s choice  = 1/3 Quickest scheme  = 1/2Quick scheme  = 0Fromm scheme (optimal) Leonard (1979)  = 1 Lax-Wendroff  – 2  = –1 Warming & Beam  – scheme (continued) Method of lines : (yet another idea for discretization.) In the  – scheme (and the linear scheme we have seen) a dependence on the time step  is included in the Courant number  To avoid this, one discretizes PDE along spatial direction first as For the FD operator L h, choose e.g.  – scheme, then apply ODE integration scheme such as RK4.

16. Monotonicity preservation of a linear advection equation Definition: (Monotonicity preserving scheme) A numerical scheme is called monotonicity preserving if for every non- increasing (decreasing ) initial data the numerical solution is non-increasing (decreasing). A linear advection equation preserves monotonicity i.e. if f(0,x): monotonic ) f(t,x): monotonic, since its general solution is. Consider a finite difference scheme that generates numerical approximation to. : data at the time step n.

17. For the uniform grid and the constant time step the (explicit or implicit) one-step scheme, in which at the (n+1)th step is uniquely determined from at the nth step, is written Theorem: (Godunov: Monotonicity preservation) The above one-step scheme is monotonicity preserving if and only if Godunov’s thorem Theorem: (Godunov’s order barrier theorem) Linear one-step second-order accurate numerical schemes for the convection equation cannot be monotonicity preserving, unless Remarks: If the numerical scheme keeps the monotonicity, a numerical solution do not shows (unphysical) oscillations (such as at the discontinuity). In these theorems, the stencils c m for the one-step FD formula are assumed to be the same at all grid points (Linear scheme). Practically, one can not have the 2 nd order linear one-step scheme.

18. For the linear s-step multi-step scheme, the same Godunov’s theorems holds. Godunov’s thorem (continued) cf.) Local truncation error of the linear one step scheme,

19. From the local truncatoin error formula, Two 2 nd order schemes: Lax-Wendroff and Warming & Beam schemes the 2 nd order scheme needs to satisfy Choice of 3 grid points j – 1, j, j +1, (m = – 1, 0, +1) results in Lax-Wendroff. Choice of 3 grid points j – 2, j – 1, j, (m = –2, –1, 0) results in Warming & Beam. ( Explicit Euler + Diffusion term centered at j.) ( 1 st order upwind + Diffusion term centered at j-1.) Writing these in the flux form Lax-Wendroff: Warming & Beam:

20. Total variation of a function TV(f) is defined by which is independent of t, hence f(t,x) is TVD. Total variation diminishing (TVD) property. For f(t,x) a solution to,, we have Definition: (TVD). If TV(f) does not increase in time, f(t,x) is called total variation diminishing or TVD. This motivates to derive a numerical scheme whose total variation of a solution Definition: A numerical scheme with this property is called TVD scheme. Theorem: (TVD property) The scheme is TVD if and only if does not increase in time step, Corollary: TVD scheme is monotonicity preserving.

21. Monotonicity preserving scheme with flux limiter function. (Flux limted schemes) Godunov’s theorem does not allow the 2 nd order linear one-step scheme. Conditions to be satisfied by the 1 st order monotonicity preserving scheme are Considering that the number of grid points for the 1 st order scheme are 2 points, resulting scheme is the 1 st -order upwind. Lax-Wendroff : 1 st order upwind ( c > 0 ): Lax-Wendroff scheme is understood as modifying the flux of 1 st order upwind. Consider a non-linear scheme that modify the flux with a limiter function (The value of differs at each cell boundary.)

22. Condition for the flux with a limiter function to be monotonicity preserving. Derive sufficient condition for the scheme with the flux to be the monotonicity preserving. Substituting the flux in the scheme, Sufficient condition for the scheme to be monotonic is This is satisfied if the flux limiter function satisfies Let the flux limiter to be a function of the slope ratio

23. Sufficient region for to have monotonicity preserving scheme. 0 2 1 Lax-Wendroff: B = 1 Warming Beam: B = r 1 White region in the right panel for and B=0 line for are allowed. Lax-Wendroff: Warming & Beam: 1 st order upwind: If (i.e. the flow is not monotonic at r j ) ) ) 1 st order upwind. If, many choices. It is desirable to have 2 nd order scheme for a smooth flow around r j = 1.

24. Minmod limiter and Superbee limiter and high resolution scheme. 0 2 1 Lax-Wendroff: B = 1 Warming Beam: B = r 1 Minmod limiter Superbee limiter is called the flux limiter function, or the slope limiter function. Minmod and Superbee are two representative limiters. 0 2 1 1 Sweby (1985) showed that the admissible limiter regions for the 2 nd order TVD scheme are those bounded by these two limiters. Schemes that is 2 nd order in the smooth flow region, and do not oscillate at the discontinuity is called high resolution scheme. TVD

25. Next steps. Numerical schemes for the system equations – ex) the Euler system Including – characteristics, shocks and Rankine-Hugoniot conditions. application of various numerical schemes approximate Riemann solver, (Godunov scheme, Roe scheme) High resolution schemes (MUSCL) Numerical schemes for the conservation laws (non-linear PDE). Including – understand characteristics introduction of weak solutions and shocks. introduction of monotonicity and TVD property. conservative form of FD schemes. application of various numerical schemes (linear schemes, Godunov scheme, high resolution schemes (MUSCL), artificial viscosity etc.) Discretization in higher dimension and general domain.