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/*
* Copyright 2014-2015 Anton Khirnov <anton@khirnov.net>
*
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
/**
* @file
* definitions of the basis functions for the spectral method
*/
#include <math.h>
#include <stdlib.h>
#include "basis.h"
#include "internal.h"
/* a set of basis functions */
typedef struct BasisSet {
/* evaluate the idx-th basis function at the specified point*/
double (*eval) (const BasisSetContext *s, double coord, int idx);
/* evaluate the first derivative of the idx-th basis function at the specified point*/
double (*eval_diff1)(const BasisSetContext *s, double coord, int idx);
/* evaluate the second derivative of the idx-th basis function at the specified point*/
double (*eval_diff2)(const BasisSetContext *s, double coord, int idx);
void (*eval_multi)(const BasisSetContext *s, double coord,
double **out, int order_start, int order_end);
/**
* Get the idx-th collocation point for the specified order.
* idx runs from 0 to order - 1 (inclusive)
*/
double (*colloc_point)(const BasisSetContext *s, int order, int idx);
} BasisSet;
struct BasisSetContext {
const struct BasisSet *bs;
double sf;
};
static double sb_even_eval(const BasisSetContext *s, double coord, int idx)
{
double val = atan2(s->sf, coord);
idx *= 2; // even only
return sin((idx + 1) * val);
}
static double sb_even_eval_diff1(const BasisSetContext *s, double coord, int idx)
{
const double sf = s->sf;
double val = atan2(sf, coord);
idx *= 2; // even only
return - sf * (idx + 1) * cos((idx + 1) * val) / (SQR(sf) + SQR(coord));
}
static double sb_even_eval_diff2(const BasisSetContext *s, double coord, int idx)
{
const double sf = s->sf;
double val = atan2(sf, coord);
idx *= 2; // even only
return sf * (idx + 1) * (2 * coord * cos((idx + 1) * val) - sf * (idx + 1) * sin((idx + 1) * val)) / SQR(SQR(sf) + SQR(coord));
}
static void sb_eval_multi(const BasisSetContext *s, double coord,
double **out, int order_start, int order_end)
{
const double sf = s->sf;
const double val = atan2(sf, coord);
for (int i = order_start; i < order_end; i++) {
const int idx = 2 * i;
const double sval = sin((idx + 1) * val);
const double cval = cos((idx + 1) * val);
out[0][i] = sval;
out[1][i] = - sf * (idx + 1) * cval / (SQR(sf) + SQR(coord));
out[2][i] = sf * (idx + 1) * (2 * coord * cval - sf * (idx + 1) * sval) / SQR(SQR(sf) + SQR(coord));
}
}
static double sb_even_colloc_point(const BasisSetContext *s, int order, int idx)
{
double t;
idx = order - idx - 1;
t = (idx + 2) * M_PI / (2 * order + 3);
return s->sf / tan(t);
}
/*
* The basis of even (n = 2 * idx) SB functions (Boyd 2000, Ch 17.9)
* SB(x, n) = sin((n + 1) arccot(|x| / L))
* They are symmetric wrt origin and decay as odd powers of x in infinity.
*/
static const BasisSet sb_even_basis = {
.eval = sb_even_eval,
.eval_diff1 = sb_even_eval_diff1,
.eval_diff2 = sb_even_eval_diff2,
.eval_multi = sb_eval_multi,
.colloc_point = sb_even_colloc_point,
};
static double cos_even_eval(const BasisSetContext *s, double coord, int idx)
{
return cos(2 * idx * coord);
}
static double cos_even_eval_diff1(const BasisSetContext *s, double coord, int idx)
{
return -2 * idx * sin(2 * idx * coord);
}
static double cos_even_eval_diff2(const BasisSetContext *s, double coord, int idx)
{
return -4 * SQR(idx) * cos(2 * idx * coord);
}
static void cos_even_eval_multi(const BasisSetContext *s, double coord,
double **out, int order_start, int order_end)
{
for (int i = order_start; i < order_end; i++) {
const double sval = sin(2 * i * coord);
const double cval = cos(2 * i * coord);
out[0][i] = cval;
out[1][i] = -2 * i * sval;
out[2][i] = -4 * SQR(i) * cval;
}
}
static double cos_even_colloc_point(const BasisSetContext *s, int order, int idx)
{
return M_PI * idx / (2 * order);
}
static const BasisSet cos_even_basis = {
.eval = cos_even_eval,
.eval_diff1 = cos_even_eval_diff1,
.eval_diff2 = cos_even_eval_diff2,
.eval_multi = cos_even_eval_multi,
.colloc_point = cos_even_colloc_point,
};
double bdi_basis_eval(BasisSetContext *s, enum BSEvalType type, double coord, int order)
{
double (*eval)(const BasisSetContext *s, double coord, int order);
switch (type) {
case BS_EVAL_TYPE_VALUE: eval = s->bs->eval; break;
case BS_EVAL_TYPE_DIFF1: eval = s->bs->eval_diff1; break;
case BS_EVAL_TYPE_DIFF2: eval = s->bs->eval_diff2; break;
}
return eval(s, coord, order);
}
double bdi_basis_eval_multi(BasisSetContext *s, double coord, double **out,
int order_start, int order_end)
{
if (s->bs->eval_multi)
s->bs->eval_multi(s, coord, out, order_start, order_end);
else {
for (int i = order_start; i < order_end; i++) {
out[0][i] = s->bs->eval(s, coord, i);
out[1][i] = s->bs->eval_diff1(s, coord, i);
out[2][i] = s->bs->eval_diff2(s, coord, i);
}
}
}
double bdi_basis_colloc_point(BasisSetContext *s, int idx, int order)
{
return s->bs->colloc_point(s, order, idx);
}
void bdi_basis_free(BasisSetContext **ps)
{
BasisSetContext *s = *ps;
free(s);
*ps = NULL;
}
BasisSetContext *bdi_basis_init(enum BasisFamily family, double sf)
{
BasisSetContext *s = calloc(1, sizeof(*s));
if (!s)
return NULL;
switch (family) {
case BASIS_FAMILY_SB_EVEN:
s->bs = &sb_even_basis;
s->sf = sf;
break;
case BASIS_FAMILY_COS_EVEN:
s->bs = &cos_even_basis;
break;
default:
free(s);
return NULL;
}
return s;
}
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