ECE539 - Advanced Theory of Semiconductors and Semiconductor Devices. Numerical Methods and Simulation / Umberto Ravaioli

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1 ECE539 - Advanced Theory of Semiconductors and Semiconductor Devices 1 General concepts Numerical Methods and Simulation / Umberto Ravaioli Introduction to the Numerical Solution of Partial Differential Equations The mathematical models of most physical processes are based on partial differential equations (PDE). Frequently, a direct analytic solution is possible only for simple cases or under very restrictive assumptions. For the solution of detailed and realistic models, numerical methods are often the only alternative available. The goal of this chapter is to introduce a number of classical examples, to familiarize the reader with numerical and practical issues in the solution of PDE s. The main objective of a numerical method is to solve a PDE on a discrete set of points of the solution domain, called discretization. In order to do so, the solution domain is divided into subdomains having the discretization points as vertices. The distance between two adjacent vertices is the mesh size. Time is also subdivided into discrete intervals, and we call timestep the interval between two consecutive times at which the solution is obtained. The PDE is then approximated, or discretized, to obtain a system of algebraic equations, where the unknowns are the solution values at the discretization points. This system of algebraic equations can then be solved on a computer by direct or iterative techniques. It is important to realize that the discretization step replaces the original equation with a new one, and that even an exact solution of the discretized problem will yield an approximate solution of the original PDE, since we introduced a discretization error. A wide variety of equations are encountered in the study of physical phenomena. Equations have been classified according to the type of phenomena involved, or according to mathematical features. It is important to attempt a classification, because the choice of a solution approach depends on the structure of the equation. Typical examples of time-dependent physical processes, given here in 1-D form, are for instance Diffusion u/ t = a u/ x Advection or drift u/ t = a u/ x Wave propagation u/ t = a u/ x For the solution of time-dependent problems of this type (initial value problems), boundary conditions on the solution domain, as well as the initial condition u(t = 0) are needed. Steady-state phenomena, characterized by zero time derivatives, can be studied by following the time-dependent process from an initial condition until a steady-state is reached. In alternative, the steady-state equations may be solved directly as a boundary value problem, which only requires the knowledge of boundary conditions. By definition, at steady-state the memory of a specific initial condition is lost. A typical boundary value problem is Laplace s equation u = 0 which could be viewed also as the steady-state of a diffusion process. particular case of Laplace s equation is a Poisson s equation u = ρ/. 1

2 The classical mathematical classification of elementary PDE s stems from the analysis of the general second order PDE α u x + β u x y + γ u y + δ u x + u y + + ηu = φ (1) The nature of the equation is determined by the coefficients, according to the following classification a) β 4αγ < 0 Elliptic (example: Poisson s equation) b) β 4αγ = 0 Parabolic (example: Diffusion equation) c) β 4αγ > 0 Hyperbolic (example: Wave equation) Many equations of practical importance may be of a mixed type, or not easily identifiable according to one of the above categories. Nevertheless, the distinction between elliptic, parabolic, and hyperbolic equations provides a very useful guideline for the selection of solution procedures. There are many approaches which are used for the discretization of the original PDE to obtain a numerical problem. The most important discretization approaches can be classified as Finite Differences Finite Elements Spectral Methods After discretization, it is necessary to check if the approximation is appropriate or if the discretized model can produce a solution at all when programmed into a computer code. For a successful solution, the numerical scheme must be stable, convergent and consistent. The scheme is stable if the solution stays bounded during the solution procedure. The scheme is convergent if the numerical solution tends to the real solution as the mesh size and the timestep tend to zero. The scheme is consistent if the truncation error tends to zero as the mesh size and the timestep tend to zero. If a numerical scheme is consistent, then stability is a necessary and sufficient condition to achieve convergence. A scheme which is stable but not consistent may converge to a solution of a different equation (with which it is consistent). A number of errors are introduced when a PDE is discretized and solved numerically. To summarize, we have Truncation error - the error introduced by the finite approximation of the derivatives. Discretization error - the error in the solution due to the replacement of the continuous equation with a discretized one. Round-off error - the computational error introduced by digital algorithms, due to the finite number of digits used in the numerical representation. The round-off error is random in nature, and normally increases when the mesh size is decreased. Conversely, the discretization error decreases with the mesh size, since more mesh points (i.e. more resolution) are introduced.

3 1-D Finite Differences The finite difference approach is the most popular discretization technique, owing to its simplicity. Finite difference approximations of derivatives are obtained by using truncated Taylor series. Consider the following Taylor expansions u(x + x) = u(x) + x u x + x u x + x3 3 u 6 x 3 + O( x4 ) () u(x x) = u(x) x u x + x u x x3 3 u 6 x 3 + O( x4 ) (3) The first order derivative is given by the following approximations: From (): Forward Difference From (3): Backward Difference u u(x + x) u(x) = + O( x) (4) x x u u(x) u(x x) = + O( x) (5) x x By substracting () (3) : Centered Difference u u(x + x) u(x x) = + O( x ) (6) x x An approximation for the second order derivative is obtained by adding () + (3): u u(x + x) u(x) + u(x x) = x x + O( x ) (7) The terms O( x) and O( x ) indicate the remainders which are truncated (truncation error) to obtain the approximate derivatives. The centered difference approximation given by (6) is more precise than the forward difference (4) or the backward difference (5) because the truncation error is of higher order, a consequence of cancellation of terms of the expansions when taking the difference between () and (3). Since the centered difference involves both neighboring points, there is more balanced information on the local behavior of the function. If a better approximation is needed, one could either reduce the size of the mesh or add more information by including higher order neighbors. It is a good exercise to derive an expression for the second order derivative including five points instead of three. Besides () and (3) we also need the expansions for u(x + x) and u(x x): u(x + x) = u(x) + x u x + 4 x u x + 8 x3 3 u 6 x x4 4 u 4 x x5 5 u 10 x 5 + O( x6 ) (8) u(x x) = u(x) x u x + 4 x u x 8 x3 3 u 6 x x4 4 u 4 x 4 3 x5 5 u 10 x 5 + O( x6 ) (9) The final result is an approximation of order O( x 4 ) u u(x + x) + 16u(x + x) 30u(x) + 16u(x x) u(x x) x 1 x (10) 3

4 By including more discretization points, it is also possible to improve the approximation for the first order derivative. The following are two possible approximations of order O( x ) u u(x + x) + 4u(x + x) 3u(x) x x (11) u u(x + x) + 8u(x + x) 8u(x x) + u(x x) (1) x 1 x Equation (11) is an improvement of the forward difference and a symmetrical expression can be derived for the backward difference. Equation (1) is an improvement of the centered difference. In order to simplify the notation, when convenient we will label the discretization points with appropriate indices. With x = h, Eq. (7) becomes u u(i + 1) u(i) + u(i 1) x h (13) Until now, we have assumed a regular grid with uniform mesh intervals, for the derivation of finite difference approximations. In many cases it may be necessary to discretize the equations on an irregular grid. Consider a non-uniform 1-D discretization, for example. If finite difference approximations are sought at the grid point i, we can use the following Taylor expansions: u u(i + 1) = u(i) + h i x + h i u x + h3 i 3 u 6 x 3 + O(h4 i ) (14) u u(i 1) = u(i) h i 1 x + h i 1 u x h3 i 1 3 u 6 x 3 + O(h4 i 1) (15) The forward and backward difference approximation for the first order derivative can be assembled as before. An expression which involves both u(i 1) and u(i + 1) is obtained from (14) (15) as u u(i + 1) u(i 1) (16) x h i + h i 1 Notice that the terms corresponding to the second order derivatives in the Taylor expansions do not cancel exactly, as in the case of the centered difference in (6) due to the nonuniformity of the mesh intervals, therefore we have a larger truncation error. It is often convenient to involve the mid-points of the mesh intervals, indicated as i 1 and i + 1, to express first order derivatives, as u x [u(i + 1 ) u(i 1 )]/(h i + h i 1 ) (17) This result can be useful to obtain an approximation for the second order derivative on the nonuniform mesh. We can write We can then express u/ x i+ 1 and obtain u x = x ( u x ) [ u x i+ 1 u x i 1 ]/( h i + h i 1 ) (18) and u/ x i 1 u x [u i+1 u i h i with centered differences, which are of order O(h ), u i u i 1 h i 1 ]( h i + h i 1 ) 1 (19) 4

5 3 1-D Examples 3.1 Finite Differences for Poisson s equation In 1-D, Poisson s equation is a simple ordinary differential equation. Assuming a uniform dielectric permittivity, we can write d V (x) dx = ρ(x) (0) where V (x) is the unknown electrical potential and ρ(x) is the space dependent charge density. The equation must be solved on a 1-D domain, where the charge density ρ(x) is sampled at the discretization points. In order to specify correctly the problem, we need two boundary conditions, since we are dealing with a second order differential equation. Such boundary conditions could be constant values for the potential at the boundary points x = 0 and x = L, or one value for the electric potential V (x) and one value for the electric field E(x), at either boundary point and in any combination. Remember that the electric field is defined as V (x) E(x) = (1) x It is trivial to show that in a 1-D problem one cannot specify the electric field at the two boundaries, since in doing so the information on the potential drop across the structure is lost. By using the finite difference approximation (7), we obtain the discretized 1-D Poisson equation V (i 1) V (i) + V (i + 1) x = ρ(i) () Let s consider first the case of two potential boundary conditions, e.g. V (1) = V 1 and V (N) = V N. The domain is divided into N 1 intervals corresponding to N grid points, of whic are boundary points, where the solution is known, and N are internal points, where the solution is unknown. In order to solve the problem, we need an additional N equations, besides the boundary conditions. Therefore, Eq. () must be written for each internal boundary point. The discretization of the differential equation (0) yields a system of N equations in N unknowns, reduced to N unknowns by eliminating the boundary conditions. In order to perform computer solutions, the system of equations must be represented in matrix form as A V = B (3) For the purpose of illustration, let s consider a uniform mesh with N = 5. equations, including boundary conditions are The discretized V (1) = V 1 V (1) V () + V (3) = x ρ() V () V (3) + V (4) = x ρ(3) V (3) V (4) + V (5) = x ρ(4) V (5) = V N This system of equations corresponds to the matrix equation 5

6 V (1) V () V (3) V (4) V (5) = V 1 ( x /)ρ() ( x /)ρ(3) ( x /)ρ(4) V N Since the boundary conditions correspond to two trivial equations, the variables V (1) and V (5) can be eliminated. The result is the following matrix equation for the internal points V () V (3) V (4) = ( x /)ρ() V 1 ( x /)ρ(3) ( x /)ρ(4) V N (4) (5) The discretization of Poisson s equation yields a tridiagonal matrix, where only three diagonals are non-zero. Matrix equations of this type can be solved quite efficiently, using a fairly compact variant of Gaussian elimination (Appendix 1). In many cases, the knowledge of the electric field is also required. By finite differencing the results for the potential, one can calculate the values of the field on the mesh points using forward, backward, and centered differences, respectively as V (i + 1) V (i) E(i) = x V (i) V (i 1) E(i) = x V (i 1) V (i + 1) E(i) = x We know by now that (8) is of order O( x ) accurate, as opposed to O( x) for (6) and (7). To compute the electric field on a boundary, only (6) and (7) can be used, since the value of the potential beyond the boundaries is necessary to evaluate the centered difference. In case the value of the electric field is needed for the mid-points of the mesh intervals, one can use the second order accurate expressions E(i + 1 (i + 1) V (i) ) = V (9) x E(i 1 (i) V (i 1) ) = V (30) x The 1-D problem can also be solved directly in terms of the electric field. Poisson s equation is substituted by Gauss law E x = ρ(x) (31) This is a first order differential equation, and only one boundary condition can be imposed. The information on the applied potential must be supplied by a condition obtained by integrating (1) over the domain, which yields (6) (7) (8) In discretized form, (31) becomes L 0 E(x)dx = V (1) V (N) (3) 6

7 ρ(i 1) x E(i) E(i 1) = (33) where ρ(i 1) now represents the average charge density inside the mesh interval i 1. We are constructing a histogram of the charge contained inside the mesh intervals, and from (33) we obtain the variation of the electric field from mesh to mesh. If we define we can rewrite (3) as E(x) = E(x) E(1) (34) L 0 E(x)dx = L On the mesh points we obtain the discretized equation 0 E(x)dx + E(1)L = V (1) V (N) (35) ρ(i 1) x E(i) E(i 1) = (36) with the boundary condition E(1) = 0. No matrix inversion is needed, because (36) is an explicit recursion. Once the distribution of E(x) is known, E(1) can be obtained from (35) by performing a numerical integration. Using Simpson s formula for the integration (there are of course many other suitable approaches) we have on a uniform mesh E(1) = V (1) V (N) L x [ E(1) + E(N) + E(i) + 4 ] E(i) 3L odd i even i where the summations are extended to internal mesh points. The distribution of the electric field on the mesh points is then given by (37) E(i) = E(i) + E(1) (38) Note that the term [V (1) V (N)]/L in (37) corresponds to the field solution when the charge is ρ = 0 throughout the domain. An extension of this method to nonuniform mesh simply requires the use of a different integration procedure in (37) Going back to Poisson equation, we mentioned above that one boundary condition for the potential and one boundary condition for the electric field can be specified at either end of the domain. Let s assume that we know the conditions E(1) = 0 and V (N) = V N. We need to choose an appropriate way to discretize (1) at x = 0, for instance E(x) = V (x) x V () V (1) x V (x) V () V (0) E(x) = (40) x x From (39), the boundary condition E(1) = 0 simply yields V (1) = V (). In the example with N = 5, we obtain now the matrix equation V (1) V () V (3) V (4) V (5) 7 = 0 ( x /)ρ() ( x /)ρ(3) ( x /)ρ(4) V N (39) (41)

8 If the centered difference (40) is choosen, the external point x = x, with label i = 0 is also involved in the boundary condition, and we have V (0) = V (). The potential V (1) at x = 0 is now solved for and we need to add an equation for the boundary point V (0) V (1) V () V (3) V (4) V (5) = 3. Finite Differences for 1-D Parabolic Equations We consider here the 1-D diffusion equation 0 ( x /)ρ(1) ( x /)ρ() ( x /)ρ(3) ( x /)ρ(4) V N u t = a u x (43) which is discretized in space and time with uniform mesh intervals x and timestep t. A simple approach is to discretize the time derivative with a forward difference as u u(i; n + 1) u(i; n) (44) t t The solution is known at time t n and a new solution must be found at time t n+1 = t n + t. Starting from the initial condition at t 1 = 0, the time evoultion is constructed after each timestep either explicitly, by direct evaluation of an expression obtained from the discretized equation, or implicitly, when solution of a system of equations is necessary. An explicit approach is readily obtained by substituting the space derivative with the 3-point finite difference evaluated at the current timestep. The algorithm, written for a generic point x i of the discretization, is u(i; n + 1) = u(i; n) + a t [ u(i 1; n) u(i; n) + u(i + 1; n) ] (45) x A fairly general implicit scheme is obtained by discretizing the space derivative with a weighted average of the finite difference approximation at t n and t n+1 (4) u(i; n + 1) λ a t [ u(i 1; n + 1) u(i; n + 1) + u(i + 1; n + 1) ] = x = u(i; n) + (1 λ) a t x [ u(i 1; n) u(i; n) + u(i + 1; n) ] (46) When λ = 1, the scheme is fully implicit. The classic Crank-Nicholson scheme is obtained when λ = 0.5 and when λ = 0 the explicit scheme in (45) is recovered. In order to cast the algorithm in matrix form, all the terms evaluated at time t n+1 are positioned on the left hand side of (46), while the terms evaluated at time t n are on the right hand side. Let s consider again a simple example with N = 5 mesh points and assume that the boundary conditions u(1; n) = u(1; n+1) = u 1 and u(5; n) = u(5; n+1) = u 5 are known. The matrix equation for the discretized diffusion equation is then 8

9 λr 1 + λr λr λr 1 + λr λr λr 1 + λr λr u(1; n + 1) u(; n + 1) u(3; n + 1) u(4; n + 1) u(5; n + 1) = u 1 RHS(; n) RHS(3; n) RHS(4; n) u 5 where RHS(i; n) represents the right hand side of (46) and r = a t / x. Other schemes originate from an evaluation of the time derivative by a centered difference approximation, in the attempt to reduce the truncation error u(i; n + 1) = u(i; n 1) + a t x [ u(i 1; n) u(i; n) + u(i + 1; n) ] (48) This scheme involves three time levels at each step of the iteration, but is explicit and is potentially much more accurate than the schemes above because of the more precise evaluation of the time derivative. Unfortunately, better truncation error is not a guarantee for the overall stability of the scheme. A variant of (48) substitutes u(i; n) with its time average [ u(i; n 1) + u(i; n + 1) ]/: (47) u(i; n + 1) = u(i; n 1) + a t x [u(i 1; n) u(i; n 1) u(i; n + 1) + u(i + 1; n)] (49) In the next section, the limits of applicability of these discretization schemes are analysed, in order to determine their validity and usefulness. It should be noted that three-level schemes need the knowledge of the solution at two consecutive times t n 1 and t n to obtain the solution at t n+1. More memory is required as well a special starting procedure. 3.3 Stability analysis A numerical scheme is unstable when, during the solution procedure, the result becomes unbounded eventually reaching overflow, independently of the choice of the computing system. The phenomenon of instability arises when the numerical procedure itself tends to amplify errors (e.g. round-off error, which is unavoidable on any digital computer) to the point that the amplified error obliterates the solution itself. This may happen even after derivatives are replaced by approximations which are more accurate from a numerical point of view. We examine here an initial value problem of the type u = L u (50) t where L is a differential space operator. The equation is solved at discrete times separated by a timestep t. Discretization of (50) will yield a matrix equation of the type A ū(t = (n + 1) t) = B ū(t = n t) (51) The solution at a given time is related to the solution at a previous timestep through a transformation of the type ū(t = (n + 1) t) = T ū(t = n t) (5) where ū is the vector of solutions obtained on the discretization points, and T = A 1 B is a transformation matrix which governs the time evolutions, according to the discretized numerical scheme. Starting from the initial condition, the iteration evolves in the following way 9

10 ū( t) = T ū(0) ū( t) = T ū( t) = T ū(0) ū(3 t) = T ū( t) = T 3 ū(0) (53) ū((n + 1) t) = T ū(n t) = T n ū(0) The eigenvalues λ i and the eigenvectors v i of the matrix T are related through T v i = λ i v i (54) Note that there are as many eigenvalues and eigenvectors as internal mesh points in the discretized domain. The solution at a given timestep can be expressed as a linear combination of the eigenvectors. At t = 0 we have ū(0) = i c n v i (55) with i = 1, N being N the total number of internal mesh points. The last equation in (53) can be rewritten as ū[(n + 1) t] = T n ū(0) = i c i (λ i ) n v i (56) The solution stays bounded if the modulus of the largest eigenvalue λ i max (spectral radius) of the matrix T satisfies the condition λ i max 1. In many cases of practical interest, stability analysis can be performed with a Fourier method. Fourier analysis is performed by expanding the solution in space Fourier components over the discretization. Then we can just study the evolution of a generic Fourier component (mode) between timesteps, since modes are orthogonal and independent of each other. A Fourier mode is expressed as F ω (i; n) = W (n) exp ( jω x i ) (57) where j is the imaginary unit, Ω is the Fourier mode wavenumber, and W (n) is the mode amplitude. The indices i and n indicate mesh point and timestep, respectively. Stability is analyzed by replacing the variable u with the generic Fourier component in the discretized scheme. The discretization is stable if the mode amplitude remains bounded at any iteration step. This is verified if W (n + 1) W (n) 1 + O( t) (58) The Fourier analysis does not include the effect of boundaries, which may affect the actual stability behavior. Boundary conditions can be introduced as additional equations when the eigenvalues of the time-evolution matrix are analyzed. 10

11 3.4 Stability analysis for the diffusion equation (a) Explicit schemes Consider (45) and substitute u(i; n) with the Fourier mode F ω (i; n) Since exp( jωx i±1 ) = W (n + 1) exp( jωx i ) = W (n) exp( jωx i ) + a t x W (n) [ exp( jωx i+1) exp( jωx i ) + exp( jωx i 1 ) ] (59) exp( jωx i ) exp(±jω x), (59) becomes W (n + 1) W (n) = 1 + a t x [ exp( jω x) + exp( jω x) ] (60) W (n + 1) W (n) = 1 4a t x sin (Ω x/) (61) For stability, we must have W (n + 1)/W (n) 1. This inequality is satisfied only when 4a t x sin (Ω x/) (6) A physical diffusion constant is a > 0, therefore the left hand side of (6) is always positive. Since 0 sin (Ω x/) 1, we can restrict ourselves to the worst case sin (Ω x/) = 1. The stability condition becomes which gives the condition for the timestep 4a t x (63) t x (64) a This result indicates that once the mesh size x is fixed, there is a limit in the choice of the timestep t. The timestep can be increased beyond this limit, only if the mesh size is also increased, leading to a reduced space resolution. The same approach can be used to analyze the stability of the three-level scheme (48) but the analysis is slightly more difficult now. For the generic Fourier mode we obtain an expression analogous to (61) W (n + 1) W (n) = W (n 1) W (n) 8a t x sin (Ω x/) (65) If we define W (n + 1)/W (n) = W (n)/w (n 1) = g, we obtain the quadratic equation with solution g = g + g 8a t x sin (Ω x/) 1 = 0 (66) 4a t x sin (Ω x/) ± a t 4 x sin 4 (Ω x/) = 0 (67) 11

12 It is easy to see that the solution with negative sign in front of the square root leads to g > 1 for any choice of finite t and x. Therefore, the scheme (48) is unconditionally unstable! The use of a more accurate approximation for the time derivative does not produce a better difference approximation for the whole equation. The modified scheme in (49) is instead always stable. However, an analysis of the local truncation error reveals that the scheme has a consistency problem. (b) Implicit schemes The Fourier stability method applied to (46) yields the equation { W (n + 1) { W (n) from which we have and for stability 1 a t x λ [ exp( jω x) + exp( jω x) ] } 1 + a t x (1 λ) [ exp( jω x) + exp( jω x) ] } W (n + 1) W (n) = 1 4 a t x (1 λ) sin ( Ω x/) a t x λ sin ( Ω x/) = (68) 1 4 a t x sin ( Ω x/) a t x λ 1 (71) sin ( Ω x/) When λ = 0 we recover the stability condition found previously for the explicit scheme (45). If we set sin (Ω x/) = 1, we have the stability condition λ ( 1 x 4 a t From this result, we can see that both the Crank-Nicholson method (λ = 0.5) and the fully implicit method (λ = 1) are unconditionally stable and any timestep t can be chosen. Of course, stability is no guarantee that the physical evolution will be properly simulated, and the timestep should be selected so that it does not exceed characteristic time constants typical of the process under investigation. 3.5 Explicit Finite Difference Methods for the Advection Equation The advection equation has the form u(x) + vu(x) = 0 (73) t x where v has the dimensions of a velocity and may be space-dependent. The advection equation is hyperbolic in nature and describes phenomena of propagation which should be solved in a natural way by obtaining the trajectories of constant solution in the space-time domain. The locus of constant solution is called characteristic. Characteristics methods for the solution are in general more expensive than finite difference methods. In particular, explicit difference methods are extremely simple to code and fast, although one has to make sure that the physics of the problem is well represented. 1 ) (69) (70) (7)

13 We can try to discretize (73) by simply using a forward difference for the time and a centered difference for the space derivatives as u(i; n + 1) = u(i; n) t [ v(i + 1) u(i + 1; n) v(i 1) u(i 1; n) ] (74) x Unfortunately, stability analysis shows that this scheme is unconditionally unstable. Some improvement is obtained by subsituting u(x i, t n ) on the right hand side with a space average t x u(i; n + 1) = u(i 1; n) + u(i + 1; n) [ v(i + 1) u(i + 1; n) v(i 1) u(i 1; n) ] (75) The scheme is now conditionally stable. Assuming for simplicity a constant velocity v, the stability condition is t x/ v. This result is usually called C.F.L. condition (from the mathematicians Courant, Friederichs and Lewy). Since x/ t has the dimensions of a velocity, the choice of discretization mesh imposes a characteristic velocity x/ t which cannot be exceeded in order to obtain physical results. There are some subtle details which are not evident from a simple stability analysis. Let s rewrite the previous schemes, adding u(i; n 1)/ u(i; n 1)/ = 0 to the left hand side of (74), and u(i; n 1) u(i; n 1) + u(i; n) u(i; n) = 0 to the left hand side of (75). Using a constant velocity, (74) becomes + u(i; n + 1) u(i; n 1) which is a second order discretization of the equation + u(i; n + 1) u(i; n) + u(i; n 1) = v t [ u(i + 1; n) u(i 1; n) ] (76) x and we obtain for (75) u(x) t + t u(x) t + v u(x) x = 0 (77) u(i; n + 1) u(i; n 1) u(i + 1; n) u(i; n) + u(i 1; n) + u(i; n + 1) u(i; n) + u(i; n 1) v t x which is in turn a second order discretization of the equation = [ u(i + 1; n) u(i 1; n) ] (78) u(x) + t u(x) t t x u(x) t x + v u(x) = 0 (79) x The stabilization in (75) is obtained by the addition of a diffusive term which is able to quench the growth of instabilities as the solution evolves in time. The introduction of this spurious diffusion causes smoothing, due to the coupling introduced between space Fourier components (modes) of the solution, which modifies the shape of the traveling solution itself. In addition, the discretization may 13

14 cause the modes to travel at different velocity (dispersion), also causing a spurious deformation of the solution. When v = x/ t, dispersion is eliminated, but obviously this holds only in the case of uniform velocity. A stable discretization of (73) is obtained with first order space differences, using a forward difference when the velocity is positive and a backward difference when the velocity is negative u(i; n + 1) = u(i; n) t x [ v(i + 1) u(i + 1; n) v(i) u(i; n) ] ; v > 0 (80) = u(i; n) + t x [ v(i) u(i; n) v(i 1) u(i 1; n) ] ; v < 0 This procedure is called upwinding, since the direction is adjusted according to the direction of the flow. A second order scheme can be obtained by using a centered time difference, which leads to a three-level time scheme, called leapfrog method u(i; n + 1) = u(i; n 1) t [ v(i + 1) u(i + 1; n) v(i 1) u(i 1; n) ] (81) x This method is stable if the C.F.L. condition is satisfied. Note that, as shown earlier, a similar method applied to the diffusion equation yields an unstable scheme. A second order scheme can be obtained considering the Taylor expansion in time u(x, t + t) = u(x, t) + t u + t u t t + O( t 3 ) (8) The time derivatives can be expressed from the advection equation itself as u(x) t = vu(x) x = v(x) u(x) x u(x) v(x) x u(x) t = v(x) [ v(x) u(x) ] x x v(x) u(x) x where we have assumed that the velocity is a slow varying space variable and that we can neglect the term u(x) v(x)/ x. The second order Lax-Wendroff scheme is then obtained by discretizing (8) (83) (84) u(i; n + 1) = u(i; n) v(i) t x [v(i + 1)u(i + 1; n) v(i 1)u(i 1; n)] (85) v(i) t x {v(i + 1/) [u(i + 1; n) u(i; n)] v(i 1/) [u(i; n) u(i 1; n)]} The last term on the right hand side of (85) corrects the instability of the scheme, and is called artificial viscosity. Note that the velocity must be known at the midpoint of the mesh intervals. 14

15 4 -D Finite Differences In the case of a 1-D domain, there are only two possible discretization approaches: uniform or nonuniform mesh intervals. A -D domain can be decomposed by using any combination of polygons having the discretization nodes as vertices. Finite difference methods are very convenient in conjunction with rectangular grids, although finite difference approximations of differential operators can be written for general meshes. Finite difference discretizations for a rectangular grid are usually assembled by considering the four neighbors of the current mesh point, although other schemes are possible. It is relatively easy to develop -D discretization on a general rectangular grid, starting from the results for the 1-D case. First order derivatives evaluated along the cartesian directions x and y are differenced in the same way, and the variables are now denoted by two indices. A finite difference approximation for the -D laplacian is easily assembled on a uniform (square) mesh, where x = y =, by combining two expressions like (7) written for the x and y direction u(i + 1, j) u(i, j) + u(i 1, j) + u = u(x, y) x + u(x, y) y = u(i, j + 1) u(i, j) + u(i, j 1) + O( ) = (86) u(i + 1, j) + u(i 1, j) 4u(i, j) + u(i, j + 1) + u(i, j 1) + O( ) This finite difference approximation involves the current point and the four nearest neighbors in the square grid (5-point formula). To improve accuracy, it is possible to involve also the second nearest neighbors (9-point formula) using the 1-D result in (10) u(x, y) x + u(x, y) y = [ u i+,j u i,j+ + 16u i+1,j + 16u i,j+1 60u i,j + 16u i 1,j + 16u i,j 1 u i,j u i,j ]/(1 ) + O( 4 ) (87) On a nonuniform rectangular grid, we can use (19) with x(i) = h i and y(j) = k j to obtain a 5-point formula which is accurate to first order u(x, y) x + u(x, y) y + [ ui+1,j u i,j h i [ ui,j+1 u i,j k j u i,j u i 1,j h i 1 u i,j u i,j 1 k j 1 ] /( h i + h i 1 ] /( k j + k j 1 ) ) (88) We can obtain the result in (88) with a more general approach called box-integration. Consider the box around the generic mesh point (x i, y j ) obtained after bisecting the mesh lines connecting the point to its neighbors. The differential operator is integrated over the volume of the box, which in -D is just the box area A box = h i 1 + h i k j 1 + k j (89) 15

16 The integral of the Laplacian over the domain D of the box can be converted into an integral over the boundary D of the domain by using Green s theorem D u(x, y) dx dy = = yj + k j yj k j 1 y j + k j y j k j 1 D u(x, y) dx dy = ( x u x i + h ) i, y dy + ( x u x i h ) i 1, y dy xi h i 1 x i + h i xi + h i x i h i 1 ( ) u u dy D x y dx ( y u x, y j + k ) j dx (90) ( y u x, y j k j 1 The derivatives at the mesh midpoints, which appear inside the line integrals in (90) can be approximated by using centered finite differences involving the neighboring mesh points. These values are used as an approximation of the average integrands for the evaluation of the line integrals, which are obtained by multiplying the average value by the corresponding length of the integration path, for instance: We then evaluate D yj + k j y j k j 1 ( x u x i + h ) i, y dy u(i + 1, j) u(i, j) h i k j 1 + k j ( u(x, y) dx dy hi 1 + h i u(i, j)a box = u(i, j) ) dx ) k j 1 + k j where we have assumed that the value of the laplacian on the mesh point is a good approximation for the average value inside the box, and we obtain finally the following finite difference approximation for the laplacian (91) (9) u(i, j + 1) u(i, j) k j [ u(i + 1, j) u(i, j) k j 1 + k j u(i, j) + h i h i 1 + h i u(i 1, j) u(i, j) k j 1 + k j + h i 1 ] ( ) h i 1 + h i hi 1 + h i k j 1 + k j / u(i, j 1) u(i, j) + k j 1 (93) It can be easily verified that the finite difference approximations (88) and (93) are identical. The box integration method provides a rigorous approximation procedure which is very useful for the discretization of complicated operators. The box integration is not limited to finite differences. Note that, up to (90), no approximation has been made and any suitable discretization method can be used from that point on D Finite Differences for Poisson s equation In the case of constant dielectric permittivity, the -D Poisson s equation is V (x, y) x + V (x, y) y = ρ(x, y) (94) 16

17 The 5-point finite difference discretization of the -D Poisson s equation is given directly by (86) for uniform square mesh V (i + 1, j) + V (i 1, j) 4V (i, j) + V (i, j + 1) + V (i, j 1) ρ(i, j) = or by (88) for nonuniform rectangular mesh (95) + [ Vi,j+1 V i,j k j [ Vi+1,j V i,j h i V i,j V i,j 1 k j 1 V i,j V i 1,j h i 1 ] ( kj + k j 1 / ] ( hi + h i 1 / ) ) ρ(i, j) = (96) The indexing in (95) and (96) assumes a natural mesh point numeration according to columns (i) and rows (j) of a rectangular grid. The scheme is often called lexicographic because the indexing follows the usual pattern of writing. An equation like (95) or (96) is written for each internal node in the -D mesh. On the boundary points we either specify a fixed potential (Dirichlet boundary condition) or the normal component of the electric field (von Neuman boundary condition). A Dirichlet boundary condition must be imposed at least at one boundary point for the problem to be specified. The ordering of the mesh points affects the structure of the discretized matrix. For illustration, let s consider a simple rectangular domain discretized with a uniform square mesh, so that x = y =, and i = 1,, 3, 4, 5 and j = 1,, 3, 4. For simplicity, fixed potential boundary conditions are imposed on all boundary points. Following the lexicographic ordering, we have equations like: Row 1: V (, 1) = V 1 V (3, 1) = V 31 V (4, 1) = V 41 Row : Row 3: V (1, ) = V 1 V (, 1) + V (1, ) 4V (, ) + V (3, ) + V (, 3) = ρ(, ) V (3, 1) + V (, ) 4V (3, ) + V (4, ) + V (3, 3) = ρ(3, ) V (4, 1) + V (3, ) 4V (4, ) + V (5, ) + V (4, 3) = ρ(4, ) V (5, ) = V 5 17

18 V (1, 3) = V 13 V (, ) + V (1, 3) 4V (, 3) + V (3, 3) + V (, 4) = ρ(, 3) V (3, ) + V (, 3) 4V (3, 3) + V (4, 3) + V (3, 4) = ρ(3, 3) V (4, ) + V (3, 3) 4V (4, 3) + V (5, 3) + V (4, 4) = ρ(4, 3) V (5, 3) = V 53 Row 4: V (, 4) = V 4 V (3, 4) = V 34 V (4, 4) = V 44 Notice that the boundary conditions for the corner points V (1, 1), V (5, 1), V (1, 4) and V (5, 4) have not been included, because they do not enter the expressions for the internal points. After elimination of the boundary conditions, the resulting matrix equations is V (, ) V (3, ) V (4, ) V (, 3) V (3, 3) V (4, 3) = ρ(,) V 1 V 1 ρ(3,) V 31 ρ(4,) V 41 V 5 ρ(,3) V 13 V 4 ρ(3,3) V 34 ρ(4,3) V 44 V 53 The discretization matrix presents five non-zero diagonals. The system of linear equations can be solved by a direct technique (e.g. Gaussian elimination) or by using an iterative procedure. 4. Iterative schemes for the -D Poisson s equation The general idea of iterative schemes is to start with an initial guess for V (i, j) and then apply repeatedly a correction scheme, which modifies the solution at each iteration, until convergence is obtained. The goal of an iterative scheme is to arrive at a solution which approximates, within a given tolerance, the exact solution of the discretized equations. Direct methods, instead, attempt an exact solution of the discretized problem, with accuracy limited only by the round-off error An iterative procedure is similar to a time-dependent problem which evolves to steady-state from an initial solution, and therefore the iteration scheme must stable and convergent. Since the actual solution cannot be exactly obtained in practice, there is the need of a stopping criterion to decide when to interrupt the iterations. At each iteration we define the error as (97) (i, j; k) = u (i, j) u(i, j; k) (98) where u (i, j) is the true solution of the discretized problem at a given mesh point (i, j) and u(i, j; k) is the solution at iteration k. A scheme is stable if the error tends to zero asymptotically. Remember that for our purposes zero is the tolerance fixed in the beginning. The error cannot in gneral reach 18

19 exactly zero because of noise introduced by round-off error. In some schemes, it is also possible for the error to increase temporarily before starting to decrease. In order to apply (98) directly, one needs to know already the solution, which is usually not the case except when the code output is compared to some known results for testing purposes. A simple way to test the convergence is to calculate for every discretized equation the residuals, obtained by replacing the solution into the discretization. For uniform discretization, we have from (95) R(i, j; k) = [u(i + 1, j; k) + u(i 1, j; k) 4u(i, j; k) + u(i, j + 1; k) +u(i, j 1; k)]/ + ρ(i, j) Obviously, the residuals tend to zero as the error (98) tends to zero, and the iterations can be stopped when the residuals are sufficiently low. In order to formulate an effective stopping criterion, the norm of local errors or residuals is used. There are various possible formulations of the norm, as defined in Hilbert spaces. We will consider here the following two norms = (i, j; k) ij 1/ (99) (100) = max{(i, j; k)} (101) Similar norms can be written for the residuals. The expression in (100) simply corresponds to the root mean square of the local error, and in the language of Hilbert spaces is also called l norm. The expression in (101) is the l norm and is obtained by searching for the maximum value. The iterative process is stopped when the selected norm is smaller than the tolerance. The l norm gives a more robust stopping criterion. If it is known that the iterative scheme is convergent, one can examine the difference between the solutions at two different steps. Convergence is assumed when the norm or u = (u(i, j; k + 1) u(i, j; k)) ij 1/ (10) u = max{u(i, j; k + 1) u(i, j; k)} (103) is smaller than the selected tolerance. This is possible because, as convergence is approached, the local variations of the iterative solution become smaller and smaller. In the following, we report a number of classical schemes for the iterative solution of the -D Poisson equation discretized with finite differences using a uniform mesh 4..1 Jacobi iteration u(i, j; k + 1) = u(i, j; k) + [u(i, j 1; k) + u(i 1, j; k) 4u(i, j; k) +u(i + 1, j; k) + u(i, j + 1; k) + ρ(i, j) /]/4 (104) 19

20 The scheme is written in a slightly redundant way, to show that the updated value is equal to the previous one plus a correction which, for this particular scheme, corresponds to one fourth of the residual calculated at the previous iteration. We already know that the residual must tend to zero as the solution is approached. The scheme in (105) can be rewritten in the compact form u(i, j; k + 1) = [u(i, j 1; k) + u(i 1, j; k) + u(i + 1, j; k) + u(i, j + 1; k) +ρ(i, j) /]/4 (105) In matrix form, we can formally write ū(k + 1) = Bū(k) + ρ (106) 4 where B is the Jacobi matrix. Note that the Jacobi iteration requires at each time step enough memory space for the solution at both iteration k and k Gauss Seidel iteration u(i, j; k + 1) = u(i, j; k) + [u(i, j 1; k + 1) + u(i 1, j; k + 1) 4u(i, j; k) +u(i + 1, j; k) + u(i, j + 1; k) + ρ(i, j) /]/4 (107) This scheme is identical to the Jacobi iteration, except that solutions at points which are scanned before (i, j) are immediately updated. From a computational point of view, this is automatically accomplished if one single array is used to store both old and updated solution. The Gauss Seidel iteration has an asymptotic convergence rate which is slightly better than the Jacobi iteration Successive Under Relaxations (SUR) u(i, j; k + 1) = u(i, j; k) + ω 4 [u(i, j 1; k) + u(i 1, j; k) 4u(i, j; k) (108) +u(i + 1, j; k) + u(i, j + 1; k) + ρ(i, j) /] The SUR scheme is a modification of the Jacobi iteration, where the correction term is multiplied by an underrelaxation parameter ω. For convergence, it must be 0 < ω 1. When ω = 1 the Jacobi iteration is recovered Successive Over Relaxations (SOR) u(i, j; k + 1) = u(i, j; k) + ω 4 [u(i, j 1; k + 1) + u(i 1, j; k + 1) 4u(i, j; k) (109) +u(i + 1, j; k) + u(i, j + 1; k) + ρ(i, j) /] 0

21 The SOR scheme is a modification of the Gauss Seidel iteration, where the correction term is multiplied by an overrelaxation parameter ω. For convergence, it must be 1 ω <. The Gauss Seidel scheme is recovered when ω = 1. The best convergence rate is obtained with an optimum relaxation parameter ω opt = 1 + (1 µ ) 1/ (110) where λ is the spectral radius (modulus of largest eigenvalue) of the Jacobi matrix B. The SOR iteration has considerably better convergence properties than the Gauss Seidel iteration. However, in the first few iterations the error actually grows, to fall sharply afterwards, while the Gauss Seidel iteration has considerable error reduction during the first few iterations Cyclic (Red-Black) SOR The grid points are subdivided into two partitions, as the red and black squares on a checkerboard. The Cyclic SOR iteration operates in two steps: Update the values on the red grid points using the SOR step u red (i, j; k + 1) = u red (i, j; k) + ω 4 [u bl(i, j 1; k) + u bl (i 1, j; k) 4u red (i, j; k) + u bl (i + 1, j; k) + u bl (i, j + 1; k) + ρ(i, j) /] (111) Update the values on the black grid points, using the SOR step u bl (i, j; k + 1) = u bl (i, j; k) + ω 4 [u red(i, j 1; k + 1) + u red (i 1, j; k + 1) 4u bl (i, j; k) + u red (i + 1, j; k + 1) + u red (i, j + 1; k + 1) + ρ(i, j) /] (11) The optimum overrelaxation parameter is the same as ω opt found for the SOR iteration. Note that at each of the two steps the partitions of red and black points are independent, since the solution is only updated on points with the same color Cyclic Chebishev iteration The Cyclic Chebishev iteration is a variation of the Cyclic SOR, where the overrelaxation parameter is varied during the iterations, according to the following scheme Iterate on red points with ω(1) = 1 Iterate on black points with ω() = (1 µ /) 1... Iterate on red points with ω(k) = (1 ω(k 1)µ /4) 1 Iterate on black points with ω(k + 1) = (1 ω(k)µ /4) 1... In the limit, ω ω opt. At the first step, the scheme behaves like the Gauss Seidel iteration, with good error reduction. 1

22 4..7 Line Iterations Line iterations are obtained by processing at one time an entire line (row) of N grid points, and the resulting implicit problem requires the solution of a system of N equations. Line iterations are possible for all the point iterations listed above. The Jacobi line iteration has the form u(i, j; k + 1) = 1 4 [u(i + 1, j; k + 1) + u(i 1, j; k + 1) (113) +u(i, j 1; k + 1) + u(i, j + 1; k) ρ(i, j) /] The scheme in (114) is similar to the point Jacobi iteration as given by (106), but here the variables corresponding to grid points on the current row are unknowns and are to be solved simultaneously at iteration (k + 1). By placing the unknowns on the left hand side, we obtain u(i 1, j; k + 1) + 4u(i, j; k + 1) u(i + 1, j; k + 1) = u(i, j 1; k) + u(i, j + 1; k) ρ(i, j) /] (114) The values at nodes on adjacent rows are evaluated at the previous iteration k. The set of equations obtained for the grid points of the row has a tridiagonal matrix, and can be solved with the method described earlier. The Gauss-Seidel line iteration is obtained simply by replacing u(i, j 1; k) with u(i, j 1; k + 1), since the latter value is already available from the iteration for row (j 1) u(i 1, j; k + 1) + 4u(i, j; k + 1) u(i + 1, j; k + 1) = u(i, j 1; k + 1) + u(i, j + 1; k) ρ(i, j) /] (115) The Successive Line Over Relaxation (SLOR) method is obtained by applying a relaxation to the Gauss Seidel line iteration u(i 1, j; k + 1) + 4u(i, j; k + 1) u(i + 1, j; k + 1) = +(1 ω)[4u(i, j; k) u(i + 1, j; k) u(i 1, j; k)] (116) +ω[u(i, j 1; k) + u(i, j 1; k + 1) ρ(i, j) /] Line iteration schemes usually converge faster and require less iterations than the corresponding point iteration schemes. The coding is somewhat more difficult, and while the point iteration requires N solution evaluations per row, the line iteration requires the solution of an N N tridiagonal matrix. The iteration cost may be comparable, provided the line iteration is coded carefully. The idea behind line iterations can be extended to block iteration, where a group (block) of adjacent rows are solved simultaneously Alternating-Direction Implicit (ADI) Iteration This iteration belongs to the category of operator splitting methods. Rather than solving the full implicit method at once, the horizontal and vertical components of the laplacian operator

23 are solved implicitly in two consecutive 1-D steps along the rows and along the columns of the grid. Since two 1-D solutions are in general not equivalent to the full -D solution, the procedure is iterated and convergence is aided by the application of a relaxation parameter β. a) Horizontal sweep u(i 1, j; k + 1/) + ( + β)u(i, j; k + 1/) u(i + 1, j; k + 1/) = u(i, j 1; k) ( β)u(i, j; k) + u(i, j + 1; k) + ρ(i, j) / (117) Every row is solved implicitly generating a tridiagonal system of equations, Similarly to the line iteration schemes. Since the values at the previous iteration are used to compute the vertical derivatives, the tridiagonal systems are all independent of each other and can be solved simultaneously. This step of the iteration resembles the Jacobi line iteration, however, here 1-D differential operators are considered for the implicit (horizontal) and the explicit (vertical) terms, and a relaxation parameter β is also used. b) Vertical sweep u(i, j 1; k + 1) + ( + β)u(i, j; k) u(i, j + 1; k + 1) = u(i 1, j; k + 1/) ( β)u(i, j; k + 1/) + u(i + 1, j; k + 1/) + ρ(i, j) / (118) Now, the same step is applied to columns of the grid, and the values at the previous k + 1/ step are used to express horizontal derivatives. Note that results obtained after the first step are very biased, and a convergence check should only be applied after both steps have been performed. If the relaxation parameter β is taken to be equal to ω opt, ADI has the same convergence rate of SOR Cyclic ADI Iteration To speed up the convergence rate of ADI, the relaxation parameter β is varied cyclically at every iteration k. For a rectangular domain, with N and M mesh intervals along the x and y sides, the values of β(k) are found according to the following algorithm a) Select s = max{m + 1, N + 1} b) Define a = 4 sin (π/s) b = 4 sin [(s 1)π/s] c = a/b δ = 3 c) Find the smallest integer m, such that δ m 1 c d) Calculate the cyclic relaxation parameters β(i) = bc ν(i) with ν(i) = (i 1)/(m 1); 1 i m When more than m iterations are necessary, the sequence of parameters β(i) is repeated. With the application of cyclical relaxation parameters, convergence is normally achieved in a much smaller number of iterations than with an equivalent SOR scheme. 3

24 4..10 Matrix Notation for the Iterative Algorithms Linear algebra notation is often used to represent in a compact form the iterative algorithms discussed above. The discretized Poisson equation can be represented by the linear system Aū = ḡ (119) where the discretization matrix A is obtained as in (97). The matrix can be decomposed as A = L + U + D (10) where D is the diagonal, and L and U are respectively the lower triangular and the upper triangular parts. The Jacobi iteration correspond to the matrix equation or Dū(k + 1) = (L + U)ū(k) + ḡ (11) ū(k + 1) = D 1 [L + U]ū(k) + D 1 ḡ (1) where D 1 (L + U) is the Jacobi matrix. In terms of the elements a i,j of the original matrix A, the Jacobi matrix for a rectangular domain with Dirichlet boundary conditions has the form D 1 (L + U) = 0 a 1 /a 11 a 13 /a a 1n /a 11 a 1 /a 0 a 3 /a... a n /a a n1 /a nn a n /a nn a n3 /a nn... 0 (13) where n = N M, with N and M the internal grid nodes on the sides of the domain. It can be shown that for this simple case the spectral radius (i.e. the modulus of the largest eigenvalue) of the Jacobi matrix is given by cos(π/m) + cos(π/n) µ = For the Gauss-Seidel method, we have the following matrix equation (14) (D + L)ū(k + 1) = Uū(k) + ḡ (15) The lower triangular matrix L is on the left hand side, because of the updating procedure discussed earlier. We can rewrite the procedure as where (D + L) 1 U is the Gauss-Seidel matrix. Suggested exercises ū(k + 1) = (D + L) 1 Uū(k) + (D + L) 1 ḡ (16) 1.1 Derive the finite difference approximations (10), (11) and(1). 1. Show that the approximation in 19 is of order O(h) where h is the local mesh. 1.3 Derive the matrix for the 1-D Poisson equation discretized with 3-point nonuniform finite differences. 4