/*
* CRANK-NICOLSON ALGORITHM 12.3
*
* To approximate the solution of the parabolic partial-differential
* equation subject to the boundary conditions
* u(0,t) = u(l,t) = 0, 0 < t < T = max t
* and the initial conditions
* u(x,0) = F(x), 0 <= x <= l:
*
* INPUT: endpoint l; maximum time T; constant ALPHA; integers m, N:
*
* OUTPUT: approximations W(I,J) to u(x(I),t(J)) for each
* I = 1,..., m-1 and J = 1,..., N.
*/
#include<stdio.h>
#include<math.h>
#define pi 4*atan(1)
#define true 1
#define false 0
double F(double X);
void INPUT(int *, double *, double *, double *, int *, int *);
void OUTPUT(double, double, int, double *, double);
main()
{
double V[25], L[25], U[25], Z[25];
double FT,FX,ALPHA,H,K,VV,T,X;
int N,M,M1,M2,N1,FLAG,I1,I,J,OK;
INPUT(&OK, &FX, &FT, &ALPHA, &N, &M);
if (OK) {
M1 = M - 1;
M2 = M - 2;
/* STEP 1 */
H = FX / M;
K = FT / N;
/* VV is used for lambda */
VV = ALPHA * ALPHA * K / ( H * H );
/* set V(M) = 0 */
V[M-1] = 0.0;
/* STEP 2 */
for (I=1; I<=M1; I++) V[I-1] = F( I * H );
/* STEP 3 */
/* STEPS 3 through 11 solve a tridiagonal linear system
using Algorithm 6.7 */
L[0] = 1.0 + VV;
U[0] = -VV / ( 2.0 * L[0] );
/* STEP 4 */
for (I=2; I<=M2; I++) {
L[I-1] = 1.0 + VV + VV * U[I-2] / 2.0;
U[I-1] = -VV / ( 2.0 * L[I-1] );
}
/* STEP 5 */
L[M1-1] = 1.0 + VV + 0.5 * VV * U[M2-1];
/* STEP 6 */
for (J=1; J<=N; J++) {
/* STEP 7 */
/* current t(j) */
T = J * K;
Z[0] = ((1.0-VV)*V[0]+VV*V[1]/2.0)/L[0];
/* STEP 8 */
for (I=2; I<=M1; I++)
Z[I-1] = ((1.0-VV)*V[I-1]+0.5*VV*(V[I]+V[I-2]+Z[I-2]))/L[I-1];
/* STEP 9 */
V[M1-1] = Z[M1-1];
/* STEP 10 */
for (I1=1; I1<=M2; I1++) {
I = M2 - I1 + 1;
V[I-1] = Z[I-1] - U[I-1] * V[I];
}
}
/* STEP 11 */
OUTPUT(FT, X, M1, V, H);
}
/* STEP 12 */
return 0;
}
/* Change F for a new problem */
double F(double X)
{
double f;
f = sin(pi * X);
return f;
}
void INPUT(int *OK, double *FX, double *FT, double *ALPHA, int *N, int *M)
{
int FLAG;
char AA;
printf("This is the Crank-Nicolson Method.\n");
printf("Has the function F(x) been created immediately\n");
printf("preceding the INPUT function? Answer Y or N.\n");
scanf("\n%c", &AA);
if ((AA == 'Y') || (AA == 'y')) {
printf("The lefthand endpoint on the X-axis is 0.\n");
*OK =false;
while (!(*OK)) {
printf("Input the righthand endpoint on the X-axis.\n");
scanf("%lf", FX);
if (*FX <= 0.0)
printf("Must be positive number.\n");
else *OK = true;
}
*OK = false;
while (!(*OK)) {
printf("Input the maximum value of the time variable T.\n");
scanf("%lf", FT);
if (*FT <= 0.0)
printf("Must be positive number.\n");
else *OK = true;
}
printf("Input the constant alpha.\n");
scanf("%lf", ALPHA);
*OK = false;
while (!(*OK)) {
printf("Input integer m = number of intervals on X-axis\n");
printf("and N = number of time intervals - separated by a blank.\n");
printf("Note that m must be 3 or larger.\n");
scanf("%d %d", M, N);
if ((*M <= 2) || (*N <= 0))
printf("Numbers are not within correct range.\n");
else *OK = true;
}
}
else {
printf("The program will end so that the function F can be created.\n");
*OK = false;
}
}
void OUTPUT(double FT, double X, int M1, double *V, double H)
{
int I, J, FLAG;
char NAME[30];
FILE *OUP;
printf("Choice of output method:\n");
printf("1. Output to screen\n");
printf("2. Output to text file\n");
printf("Please enter 1 or 2.\n");
scanf("%d", &FLAG);
if (FLAG == 2) {
printf("Input the file name in the form - drive:name.ext\n");
printf("for example: A:OUTPUT.DTA\n");
scanf("%s", NAME);
OUP = fopen(NAME, "w");
}
else OUP = stdout;
fprintf(OUP, "CRANK-NICOLSON METHOD\n\n");
fprintf(OUP, " I X(I) W(X(I),%12.6e)\n", FT);
for (I=1; I<=M1; I++) {
X = I * H;
fprintf(OUP, "%3d %11.8f %13.8f\n", I, X, V[I-1]);
}
fclose(OUP);
}