Physics Research Summer2009

Advection Non-trivial solutions to the advection equation (right-moving) (1) give a moving pulse, whose direction is determined by the sign of v. Positive v = right-moving. f(x) tx Figure 1 For arbitrary functions f, the solution to the advection equation is f(u), where u = (x - vt). Proof: This solution works for all cases. v>0 yields a right-moving wave, and v<0 a left-moving wave. Consider Figure 1, where time advances with each graph from left to right. Notice that each point on the curve is simply being dragged forward. If we mark two x values and their corresponding f(x) values, the movement of the pulse along characteristic lines (lines along which the function does not change) becomes apparent. Figure 3 is a contour plot for Figure 2 as it advances with time. t xb xax xb xa Figure 2Figure 3 Derivative Approximations Forward Time Centered Space (FTCS) (Garcia 2000) Consider a set of grid points with known x and f(x) values. We will say this set of data points exists at time level t = tn, and xi denotes x at the ith place. The distance between points is equal to Δx, and the duration between time levels is Δt. What if we want to find information at another time level? (Figure 4) fin+1? Figure 4 finknown We assume this data obeys the advection equation and estimate the function values at time level t = tn+1 using a forward time centered space approximation (FTCS). Recall equation (1). ⇒∂f∂tx=xi,t=tn=-v∂f∂xx=xi,t=tn The partial derivative on the right side of this equation can be approximated as ≈fi+1n- fi-1n2∆x The partial derivative on the left as ≈fin+1- fin∆t Equating the two with the appropriate constant v yields fin+1- fin∆t=-vfi+1n- fi-1n2∆x Solving for fin+1gives fin+1=fin+fi+1n- fi-1n2∆x-v∆t (2) The approximation of the space derivative is second order accurate by virtue of being a centered derivative approximation. The approximation of the time derivative is only first order accurate since it is a forward derivative approximation. The computational molecule, which shows what points are used to approximate the value at the desired point, for FTCS is Figure 5 FTCS is unconditionally unstable. If allowed to run for long enough, an FTCS algorithm will always blow up – infinities will start to show up in later time steps. In most numerical methods for evolving hyperbolic partial differential equations, the stability of the system is determined by the Courant factor. The courant factor is α=v∆t∆x, or the ratio of physical wave speed to the speed at which the algorithm propagates information. tConsider a physical light cone. x Figure 6 Only events within in the light cone can influence or be influenced by an observer at the origin. For our purposes, the physical light cone of the wave is determined by the physical speed of the wave. The numerical light cone is determined by the spacing of the grid on which function values are known. For example, looking at the computational molecule for FTCS, you can see that numerical light cone. Figure 7 Computationally, the only information affecting the approximation of the function value at the gray point is the information at the three black points. Consider a physical light cone mapped onto Figure 7. Figure 8 Here, the physical light cone is smaller than the numerical light cone because v<ΔxΔt. Stable approximations will have a numerical light cone bigger than a physical light cone, or a Courant factor α≤1. Keeping with the physical light cone analogy (Figure 6), the computational system can be understood heuristically as taking into account all possible physical influences. However, recall that FTCS is unconditionally unstable, and will eventually blow up even when this condition is satisfied. To get an idea of the problems with FTCS, consider a simple situation such as when a Courant factor is α=1. Here, we would expect that this algorithm would essentially drag the pulse along characteristic lines. This is not true for FTCS. Recall equation (2). fin+1=fin+fi+1n- fi-1n2∆x-v∆t For a Courant factor α=1, the algorithm becomes fin+1=fin-fi+1n- fi-1n2 (3) Since we have provided the algorithm with a very simple system and are looking at a symmetric situation that it should be able to handle easily, we should expect a rather clean-looking answer like fin+1=fi-1n (4) In the case of equation (4), the wave is simply pulled along the characteristic line, which is what should happen computationally when numerical and physical speeds are equal; however, this is clearly not the case for FTCS. Although this is not a rigorous explanation for the fact that FTCS is unconditionally unstable, it does show some of the issues with using FTCS, even when solving very simple systems. Lax (Garcia 2000) Instead of using a known data point at (xi,tn)and then adding the time derivative multiplied by ∆t(-v), the Lax method uses an average of the two points adjacent to xi at time level tnThe lax method uses the average of points to find the point at the next time step. Average Figure 11 fin+1=12fi+1n+ fi-1n+ ∆t-vfi+1n- fi-1n2∆x (7) The Lax method includes a diffusion term, which acts as though it takes energy away from the wave, preventing it from blowing up to infinity. As is expected, the Lax method becomes unstable with large Courant factors, but it will also become unstable with very small Courant factors. In the latter condition, the pulse dissipates rather quickly. Centered Time Centered Space (Garcia 2000) The Centered Time Centered Space method uses two initial time levels of data in order to get centered derivatives that are second-order accurate in both space and time n Figure 9 From before -v∂f∂xx=xi,t=tn=∂f∂tx=xi,t=tn (5) The derivative approximations are ∂f∂xx=xi,t=tn≈fi+1n-fi-1n2∆x And ∂f∂tx=xi,t=tn≈fin+1-fin-12∆t Solving equation (5) yields fin+1=fin-1+fi+1n- fi-1n∆x-v∆t (6) Note that the computational molecule for CTCS is Figure 10 If α=1, we can assume within a reasonable uncertainty that fin-1 = fi+1n. Therefore, equation (6) reduces to fin+1≈fi-1n. This becomes exact if initial data ensures that fi'=fi+12. Lax-Wendroff (Garcia 2000) This method finds second derivatives, which it then uses to get to the next time step. Figure 12 provides a graphical representation of estimating second derivatives. The derivative is being approximated at the center black point. ∂f∂x∂f∂x ∂2f∂x2 Figure 12 Consider a Taylor series expansion ft+∆t,x=f(t,x)+∆t∂f∂tt,x+∆t22!∂2f∂tt,x+… The FTCS effectively truncates the Taylor series estimation at the term with the first-order partial differential. Lax-Wendroff carries the estimation one term further in the expansion. An alternative way of using the Lax-Wendroff method takes a lax step to an intermediate time level and then “leaps” across that intermediate step to the desired value. 643Black dots = initial data21 5 Figure 12 For each step in Figure 12: ,[object Object]

∂f∂xx=xi+12,t=tn≈fi+1n-fin∆x

fi-12n+12≈fi-1n + fin2+∆t2fin- fi-1n∆x-v

fi+12n+12≈fin + fi+1n2+∆t2fi+1n- fin∆x-v

∂f∂xx=xi,t=tn+12≈fi+12n+12-fi-12n+12∆x

fin+1≈∆t(v)2∆x+∆t2v22∆x2fi-1n+1-∆t2v2∆x2fin+∆t(-v)2∆x+∆t2v22∆x2fi+1nLax-Wendroff is essentially FTCS plus a diffusion term. Although the difference between FTCS and Lax-Wendroff is basically a matter of an additional term, Lax-Wendroff is stable for the Courant condition α≤1 and is second-order accurate in both time and space. This method does suffer from a dissipation problem similar to the Lax method. However, Lax-Wendroff is not affected as greatly by this issue as Lax is. Iterated Crank-Nicholson (Teukolsky 2008) The iterated Crank-Nicholson method makes a guess at the values of the next time step and then improves the guess by averaging. This method is stable for 4n+2 and 4n+3 iterations where n = 0,1,2,3,… However, doing more than two corrections is not really helpful. Consider the advection equation (1) without the velocity factor. FTCS gives fin+1-fin∆t=fi+1n-fi-1n2∆x Backwards differencing (which is unconditionally stable) gives fin+1-fin∆t=fi+1n+1-fi-1n+12∆x (8) Crank-Nicholson averages these two methods. The first iteration of Crank-Nicholson calculates an intermediate variable using FTCS: fin+1-fin∆t=fi+1n-fi-1n2∆x (9) Another intermediate variable is found by averaging fin+12=12fin+1+fin (10) Then the time step is made using f in an FTCS approximation fin+1-fin∆t=fi+1n+12-fi-1n+122∆x (11) A two-step iteration is done the same way. After equations (9) and (10): fin+1-fin∆t=fi+1n+12-fi-1n+122∆x (12) fin+12=12fin+1+fin The time step is then made as in equation (12). fin+1-fin∆t=fi+1n+12-fi-1n+122∆x (13) As shown by Teukolsky, 2008, Crank-Nicholson is stable for courant factors α≤2, or α24≤1. Upwind Method (Garcia 2000) This method exploits the directionality of the wave by using only points that would trail the wave (physically) in the calculations. One benefit of this method is that we need only provide a boundary condition for the side of the computational domain away from which the wave travels. The computational molecule for this method is Figure 13 And its numerical light cone is Figure 14 The algorithm for a right-moving pulse, obeying the advection equation is fin=fin-1-v∆t∆xfin-1-fi-1n-1 (14) At the left boundary, some known value is provided (e.g., constant at 0) at each time step. Providing known values as boundaries is known as Drichlet boundary conditions. A right boundary condition is not necessary as the wave will simply move off the edge of the computational world – outgoing boundary condition. For a left-moving pulse, we would simply need to mirror the points used (e.g., fin becomes the left-most point for a given set rather than the right-most). Beam-Warming (Dullemond and Kuiper 2008) Whereas the upwind method only used two points at the previous time level (Figure 13), the Beam-Warming method uses three. Figure 15 The partial differential equation (PDE) for the Beam-Warming method looks something like ∂f∂t=v∂f∂xx=xi+12,t=tn-v2∂2f∂x2x=xi+1,t=tn (15) The second derivative term is a diffusion term, which acts as if it is taking energy away from the pulse, preventing it from blowing up. The right-moving algorithm for this method is fin+1=1-3v∆t2∆x+v2∆t22∆x2fin+2v∆t∆x-v2∆t2∆x2fi-1n+(-v)∆t2∆x+v2∆t22∆x2fi-1n (16) Again, these points can simply be mirrored for a left-moving algorithm. Also, note that this system requires boundary conditions for the two left-most points, whereas upwind only needed a boundary condition for the one, left-most point. Box Method The Box method uses two points at the previous time step and one point on the current time step in order to approximate a point at the current time step. in-1 Figure 16 The PDE for the Box method is -v∂f∂xx=xi-12,t=tn-1-v∂f∂xx=xi-12,t=tn2=∂f∂tx=xi,t=tn-12+∂f∂tx=xi-1,t=tn-122 (17) The algorithm is fin=fi-1n-1+fin-1-fi-1nv∆t-∆xv∆t+∆x Recall that α=v∆t∆x So, the algorithm can be reduced to fin=fi-1n-1+fin-1-fi-1n1-α1+α (18) Note the numerical light cone for this system Figure 17 Because the light cone encompasses all previous points at all previous time levels, this system should be stable even for very large Courant factors. Wave Equation The simplest one-dimensional wave equation is (19). This equation has the solution f(u), where u = x + vt or x – vt. Proof: Show that f(u) where u = x + vt is a solution to equation 2. 1st partial 2nd partial 1st partial 2nd partial The proof for u = x – vt is similar. Also, it has already been shown that characteristics have velocities ±v. Crank Nicholson and the Wave Equation Recall the wave equation (19). In two-variable form, this equation is. ∂f∂t=ft ∂ft∂t=v2∂2y∂x2 (20) We can discretize this equation using Crank-Nicholson. However, we need two pieces of initial data. Either f and ft at one time level or f at two time levels. For discretization, I will assume the first situation, with known values at time level t = tn-1. fin-fin-1∆t=ftin-1 ftin-ftin-1∆t=v2fi+1n-1-2fin-1+fi-1n-1∆x2 fin-fin-1∆t=ftin-12=ftin-1+ftin2 ftin-ftin-1∆t=v2fi+1n-12-2fin-12+fi-1n-12∆x2 Note that here fin-12=fin+fin-12 fin-fin-1∆t=ftin-12=ftin-1+ftin2 ftin-ftin-1∆t=v2fi+1n-12-2fin-12+fi-1n-12∆x2 Note that here fin-12=fin+fin-12 Boundary Conditions for the Wave Equation From the section on approximating numerical derivative, the Upwind, Beam-Warming, and Box methods are particularly useful for establishing boundary conditions. When a wave is evolved forward in time numerically, the system doing the evolution loses information at the boundaries of the domain over which the wave is being evolved. Information lost at boundariesTime advances Numerical Domain Figure 18 Consider the wave equation (19) This equation has as a solution fx,t= Fx-vt+ Gx+vt (21) The F term is the right-moving part of the wave and the G term is the left-moving part. These two parts are independently both solutions of the advection equation (1). ∂F∂t=-v∂F∂x ∂G∂t=v∂G∂x Therefore, we can use the aforementioned methods for boundary conditions by forcing the wave equation to obey the appropriate advection equation (left-moving or right-moving) at the boundaries. Wave equation in Spherical Coordinates The three dimensional wave equation in spherical coordinates is ∂2f∂t2=v2∇2f (22) The Laplacian in spherical coordinates is ∇2f=1r2∂∂rr2∂f∂r+1rrsinθ∂∂θsinθ∂f∂θ+1r2sinθ∂2f∂ϕ2 23 If f=f(r,t) ∂f∂θ=0 & ∂f∂ϕ=0 The Laplacian reduces to ∇2f(r,t)=1r2∂∂rr2∂f∂r So ∂2f∂t2=v21r2∂∂rr2∂f∂r ∂2f∂t2=v2∂2f∂r2+2r∂f∂r (24) Note that at the origin, the 2r∂f∂r term will be zero, because ∂f∂r is 0 at the origin. Boundary Conditions for the Wave Equation in 3-D Outgoing Boundary Recall that solutions to the wave equation (19) have left and right moving parts that obey the advection equation. See equation (21), copied here. fx,t= Fx-vt+ Gx+vt However, the 3-D wave equation in spherical coordinates (24) has an additional term. ∂2f∂t2=v2∂2f∂r2+2r∂f∂r How can we use our out-going boundary conditions (Upwind, Beam-Warming, and Box) with a 3-D wave equation? Letg=(r±vt) That is to say,∂g∂t=±v∂g∂r Also, letfr=hrg(r+ϵvt) Where ϵ=±1 Recalling that∂2f∂t2=v21r2∂∂rr2∂f∂r Find hr that makes this work. ∂f∂t=hg'ϵv ∂f∂r=h'g+hg' ∂2f∂t2=hg''ϵv2 ∂2f∂r2=h''g+g'h'+h'g'+hg''=h''g+2h'g'+hg'' ∂2f∂t2=v2∂2f∂r2+2r∂f∂r v2hg''=v2(h''g+2h'g'+hg''+2rh'g+2rhg' 0=h''+2rh'g+2h'+1rhg' h''= -2rh h'=-1rh Now all we have to do is solve the ODE dhdr=-hr dhh=-drr h=cr Now, ft,r=gr±vtr ∂∂trf=-v∂∂rrf Considering the previous boundary conditions, such as the box method, all that needs to change is f⇒rf. For example, the Box method algorithm (18) becomes rifin=ri-1fi-1n-1+rifin-1-ri-1 fi-1n1-α1+α In short, rf obeys a simpler wave equation. Boundary at the Origin Findf'0r At the origin limr->0f'rr Consider the Taylor series f(r)=f(0)+rf'(0)1!+r2f''(0)2!+r3f'''(0)3!+… So, we can see that limr->0f'rr=limr->0f0+rf'01!+r2f''02!+r3f'''03!+…=f''0 Exploiting the fact that the wave is symmetric about the origin, the second derivative at the origin can easily be approximated as ∂2f∂r2≈2f∆r-f0∆r2 (25) ∂f∂r∂f∂r ∂2f∂r2 Figure 19 Also, by considering the Laplacian in Cartesian coordinates ∂2f∂t2=v2∂2f∂y2+∂2f∂y2+∂2f∂z2 Due to the spherical symmetry about the origin (r=0), ∂2f∂x2=∂2f∂y2+∂2f∂z2 Therefore, ∂2f∂x2=3v2∂2f∂r2 Discretizing the Wave Equation CFL ∂2f∂t2=v2∂2f∂r2+2r∂f∂r fin+1-2fin+fin-1∆t2=v2f i-1n-2fin+fi+1n∆r2+2rfi+1n-fi-1n2∆r At r = 0 ∂2f∂t2=3v2∂2f∂22 At outer boundary ∂g∂t=-v ∂g∂r Where g=fr Crank-Nicholson ∂2f∂t2=v2∂2f∂r2+2r∂f∂r Let∂f∂t=ft ⇒ ∂ft∂t=v2∂2f∂r2+2r∂f∂r fin-fin-1∆t=ftin-1 ftin-ftin-1∆t=v2fi+1n-1-2fin-1+fi-1n-1∆x2+2rfi+1n-1-fi-1n-12∆x fin-fin-1∆t=ftin-12=ftin-1+ftin2 ftin-ftin-1∆t=v2fi+1n-12-2fin-12+fi-1n-12∆x2+2rfi+1n-12-fi-1n-122∆x fin-fin-1∆t=ftin-12=ftin-1+ftin2 ftin-ftin-1∆t=v2fi+1n-12-2fin-12+fi-1n-12∆x2+2rfi+1 n-12-fi-1n-122x2 At r = 0 ∂2f∂t2=3v2∂2f∂22 At outer boundary ∂g∂t=-v ∂g∂r Where g=fr

Physics Research Summer2009

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