1. Homepage : http://jkl.postech.ac.kr Boundary Value Problems of Ordinary Differential Equations & Scharfetter-Gummel method (Computational E&M: 490D) Sung Soo Yang, and Jae Koo Lee March 23, 2004 P lasma A pplication M odeling

2. Plasma Application Modeling @ POSTECH Boundary value problems Type of ProblemsAdvantagesDisadvantages Shooting methodAn existing program for initial value problems may be used Trial-and-error basis. Application is limited to a narrow class of problems. Solution may become unstable. Finite difference method using the tridiagonal solution No instability problem. No trial and error. Applicable to nonlinear problems with iteration. Problem may have to be developed for each particular problem. 1) : Dirichlet type boundary condition * Three types of boundary conditions 1) : Neumann type boundary condition 1) : Mixed type boundary condition

3. Plasma Application Modeling @ POSTECH Solve difference equation, With the boundary conditions,   x = 012 i = 012 910 9 Especially for i = 1, known y(0)=1 Program 10-1

4. Plasma Application Modeling @ POSTECH For i = 10, Summarizing the difference equations obtained, we write Tridiagonal matrix Program 10-1

5. Plasma Application Modeling @ POSTECH Solution Algorithm for Tridiagonal Equations (1) R2R2 R3R3 Based on Gauss elimination

6. Plasma Application Modeling @ POSTECH Solution Algorithm for Tridiagonal Equations (2)

7. Plasma Application Modeling @ POSTECH void trdg(float a[], float b[], float c[], float d[], int n) /* Tridiagonal solution */ { int i; float r; for ( i = 2; i <= n; i++ ) { r = a[i]/b[i - 1]; b[i] = b[i] - r*c[i - 1]; d[i] = d[i] - r*d[i - 1]; } d[n] = d[n]/b[n]; for ( i = n - 1; i >= 1; i-- ) { d[i] = (d[i] - c[i]*d[i + 1])/b[i]; } return; } Recurrently calculate the equations in increasing order of i until i=N is reached Calculate the solution for the last unknown by Calculate the following equation in decreasing order of i Solution Algorithm for Tridiagonal Equations (3)

8. Plasma Application Modeling @ POSTECH /* CSL/c10-1.c Linear Boundary Value Problem */ #include /* a[i], b[i], c[i], d[i] : a(i), b(i), c(i), and d(i) n: number of grid points */ int main() { int i, n; float a[20], b[20], c[20], d[20], x; void trdg(float a[], float b[], float c[], float d[], int n); /* Tridiagonal solution */ printf( "\n\nCSL/C10-1 Linear Boundary Value Problem\n" ); n = 10; /* n: Number of grid points */ for( i = 1; i <= n; i++ ) { x = i; a[i] = -2; b[i] = 5; c[i] = -2; d[i] = exp( -0.2*x ); } Program 10-1

9. Plasma Application Modeling @ POSTECH d[1] = d[1] + 2; d[n] = d[n]*0.5; b[n] = 4.5; trdg( a, b, c, d, n ); d[0] = 1; /* Setting d[0] for printing purpose */ printf( "\n Grid point no. Solution\n" ); for ( i = 0; i <= n; i++ ) { printf( " %3.1d %12.5e \n", i, d[i] ); } exit(0); } Program 10-1 CSL/C10-1 Linear Boundary Value Problem Grid point no. Solution 0 1.00000e+00 18.46449e-01 27.06756e-01 35.85282e-01 44.82043e-01 53.95162e-01 63.21921e-01 72.59044e-01 82.02390e-01 91.45983e-01 107.99187e-02

10. Plasma Application Modeling @ POSTECH Program 10-3 An eigenvalue problem of ordinary differential equation We assume a b   i-1ii+1

11. Plasma Application Modeling @ POSTECH Program 10-3 (Inverse Power method) Step1 :for all i andare set to an arbitrary initial guess. Step2 : is solved by the tridiagonal solution for Step3 : The next estimate for is calculated by the equation: Step4 : is solved by the tridiagonal solution for Step5 : The operations similar to Step 3 and 4 are repeated as the iteration cycle t increases. Step6 : The iteration is stopped when the convergence test is satisfied. Criterion for convergence

12. Plasma Application Modeling @ POSTECH Program 10-3 main() { while (TRUE) { k = 0; n = 10; ei = 1.5; it = 30; ep = 0.0001; for (i=1; i <= n; i++) { as[i] = -1.0; bs[i] = 2.0; cs[i] = -1.0; f[i] = 1.0; ds[i] = 1.0; } printf("\n It. No. Eigenvalue\n"); while (TRUE) { k = k+1; for (i=1; i<=n; i++) fb[i] = f[i]*ei; for (i=1; i<=n; i++) { a[i] = as[i]; b[i] = bs[i]; c[i] = cs[i]; d[i] = ds[i]*fb[i]; } trdg(a, b, c, d, n); sb = 0; s = 0; for (i=1; i <= n; i++) { f[i] = d[i]; s = s + f[i]*f[i]; sb = sb + f[i]*fb[i]; } eb = ei; ei = sb/s; printf("%3.1d %12.6e \n", k, ei); if (fabs(1.0-ei/eb) <= ep) break; if (k >it) { printf("Iteration limit exceeded.\n"); break; } Step 1 Step 6 Step 3 Step 2,4

13. Plasma Application Modeling @ POSTECH z = 0; for (i=1; i <= n; i++) if (fabs(z) <= fabs(f[i])) z = f[i]; for (i=1; i <= n; i++) f[i] = f[i]/z; eigen = ei; printf("Eigenvalue = %g \n", eigen); printf("\n Eigenfunction\n"); printf("i f(i)\n"); for (i=1; i<=n; i++) printf("%3.1d %12.5e \n", i, f[i]); printf("------------------------------------\n"); printf("Type 1 to continue, or0 to stop. \n"); scanf("%d", &kstop); if (kstop != 1) exit (0); } It. No. Eigenvalue 1 8.196722e-02 2 8.102574e-02 3 8.101422e-02 4 8.101406e-02 Eigenvalue = 0.0810141 Eigenfunction i f(i) 1 2.84692e-01 2 5.46290e-01 3 7.63595e-01 4 9.19018e-01 5 1.00000e+00 6 1.00000e+00 7 9.19018e-01 8 7.63594e-01 9 5.46290e-01 10 2.84692e-01 ------------------------------------ Type 1 to continue, or0 to stop. Program 10-3

14. Plasma Application Modeling @ POSTECH Scharfetter-Gummel method 2D discretized continuity eqn. integrated by the alternative direction implicit (ADI) method Tridiagonal matrix Scharfetter-Gummel method

15. Plasma Application Modeling @ POSTECH Scharfetter-Gummel method   Flux is constant between half grid points and calculated at the grid point Flux is a linear combination of and Analytic integration between i and i+1 leads to

16. Plasma Application Modeling @ POSTECH Scharfetter-Gummel method where, The main advantage of SG scheme is that is provides numerically stable estimates of the particle flux under all conditions (Big potential difference)(small potential difference)

17. Plasma Application Modeling @ POSTECH Scharfetter-Gummel method In case of Drift flux

18. Plasma Application Modeling @ POSTECH Scharfetter-Gummel method In case of Diffusion flux