1 // Copyright (c) 2017-2024, Lawrence Livermore National Security, LLC and other CEED contributors. 2 // All Rights Reserved. See the top-level LICENSE and NOTICE files for details. 3 // 4 // SPDX-License-Identifier: BSD-2-Clause 5 // 6 // This file is part of CEED: http://github.com/ceed 7 8 /// @file 9 /// Helper functions for solving the Riemann problem. 10 // The left and right states are specified from the perspective of an outward-facing normal vector pointing left to right: 11 // 12 // (domain) 13 // / (outward facing normal) 14 // |------------| / 15 // | | / 16 // | Left |----> Right 17 // | (Interior) | (Exterior) 18 // |------------| 19 // 20 // The right state is exterior to the domain and the left state is the interior to the domain. 21 // Much of the work references Eleuterio F. Toro's "Riemann Solvers and Numerical Methods for Fluid Dynamics", 2009 22 #include "newtonian_state.h" 23 #include "newtonian_types.h" 24 25 enum RiemannFluxType_ { RIEMANN_HLL, RIEMANN_HLLC }; 26 typedef enum RiemannFluxType_ RiemannFluxType; 27 28 typedef struct { 29 CeedScalar left, right; 30 } RoeWeights; 31 32 CEED_QFUNCTION_HELPER RoeWeights RoeSetup(CeedScalar rho_left, CeedScalar rho_right) { 33 CeedScalar sqrt_left = sqrt(rho_left), sqrt_right = sqrt(rho_right); 34 RoeWeights w = {sqrt_left / (sqrt_left + sqrt_right), sqrt_right / (sqrt_left + sqrt_right)}; 35 return w; 36 } 37 38 CEED_QFUNCTION_HELPER RoeWeights RoeSetup_fwd(CeedScalar rho_left, CeedScalar rho_right, CeedScalar drho_left, CeedScalar drho_right) { 39 CeedScalar sqrt_left = sqrt(rho_left), sqrt_right = sqrt(rho_right); 40 CeedScalar square_sum_root = Square(sqrt_left + sqrt_right); 41 CeedScalar r_right = (sqrt_left / (2 * sqrt_right * square_sum_root)) * drho_right - (sqrt_right / (2 * sqrt_left * square_sum_root)) * drho_left; 42 CeedScalar r_left = (sqrt_right / (2 * sqrt_left * square_sum_root)) * drho_left - (sqrt_left / (2 * sqrt_right * square_sum_root)) * drho_right; 43 RoeWeights dw = {r_left, r_right}; 44 return dw; 45 } 46 47 CEED_QFUNCTION_HELPER CeedScalar RoeAverage(RoeWeights r, CeedScalar q_left, CeedScalar q_right) { return r.left * q_left + r.right * q_right; } 48 49 CEED_QFUNCTION_HELPER CeedScalar RoeAverage_fwd(RoeWeights r, RoeWeights dr, CeedScalar q_left, CeedScalar q_right, CeedScalar dq_left, 50 CeedScalar dq_right) { 51 return q_right * dr.right + q_left * dr.left + r.right * dq_right + r.left * dq_left; 52 } 53 54 CEED_QFUNCTION_HELPER StateConservative Flux_HLL(State left, State right, StateConservative flux_left, StateConservative flux_right, 55 CeedScalar s_left, CeedScalar s_right) { 56 CeedScalar U_left[5], U_right[5], F_right[5], F_left[5], F_hll[5]; 57 UnpackState_U(left.U, U_left); 58 UnpackState_U(right.U, U_right); 59 UnpackState_U(flux_left, F_left); 60 UnpackState_U(flux_right, F_right); 61 for (int i = 0; i < 5; i++) { 62 F_hll[i] = (s_right * F_left[i] - s_left * F_right[i] + s_left * s_right * (U_right[i] - U_left[i])) / (s_right - s_left); 63 } 64 StateConservative F = { 65 F_hll[0], 66 {F_hll[1], F_hll[2], F_hll[3]}, 67 F_hll[4], 68 }; 69 return F; 70 } 71 72 CEED_QFUNCTION_HELPER StateConservative Flux_HLL_fwd(State left, State right, State dleft, State dright, StateConservative flux_left, 73 StateConservative flux_right, StateConservative dflux_left, StateConservative dflux_right, 74 CeedScalar S_l, CeedScalar S_r, CeedScalar dS_l, CeedScalar dS_r) { 75 CeedScalar U_l[5], U_r[5], F_r[5], F_l[5]; 76 UnpackState_U(left.U, U_l); 77 UnpackState_U(right.U, U_r); 78 UnpackState_U(flux_left, F_l); 79 UnpackState_U(flux_right, F_r); 80 81 CeedScalar dU_l[5], dU_r[5], dF_r[5], dF_l[5], dF_hll[5] = {0.}; 82 UnpackState_U(dleft.U, dU_l); 83 UnpackState_U(dright.U, dU_r); 84 UnpackState_U(dflux_left, dF_l); 85 UnpackState_U(dflux_right, dF_r); 86 for (int i = 0; i < 5; i++) { 87 const CeedScalar S_diff = S_r - S_l; 88 89 dF_hll[i] += (S_l * (-F_l[i] + F_r[i] + S_l * U_l[i] - S_l * U_r[i]) / Square(S_diff)) * dS_r; 90 dF_hll[i] += (S_r * (F_l[i] - F_r[i] - S_r * U_l[i] + S_r * U_r[i]) / Square(S_diff)) * dS_l; 91 dF_hll[i] += (S_r * dF_l[i] - S_l * dF_r[i] + S_r * S_l * (dU_r[i] - dU_l[i])) / S_diff; 92 } 93 StateConservative dF = { 94 dF_hll[0], 95 {dF_hll[1], dF_hll[2], dF_hll[3]}, 96 dF_hll[4], 97 }; 98 return dF; 99 } 100 101 CEED_QFUNCTION_HELPER void ComputeHLLSpeeds_Roe(NewtonianIdealGasContext gas, State left, CeedScalar u_left, State right, CeedScalar u_right, 102 CeedScalar *s_left, CeedScalar *s_right) { 103 const CeedScalar gamma = HeatCapacityRatio(gas); 104 105 RoeWeights r = RoeSetup(left.U.density, right.U.density); 106 // Speed estimate 107 // Roe average eigenvalues for left and right non-linear waves. 108 // Stability requires that these speed estimates are *at least* as fast as the physical wave speeds. 109 CeedScalar u_roe = RoeAverage(r, u_left, u_right); 110 111 CeedScalar H_left = TotalSpecificEnthalpy(gas, left); 112 CeedScalar H_right = TotalSpecificEnthalpy(gas, right); 113 CeedScalar H_roe = RoeAverage(r, H_left, H_right); 114 CeedScalar a_roe = sqrt((gamma - 1) * (H_roe - 0.5 * Square(u_roe))); 115 116 // Einfeldt (1988) justifies (and Toro's book repeats) that Roe speeds can be used here. 117 *s_left = u_roe - a_roe; 118 *s_right = u_roe + a_roe; 119 } 120 121 CEED_QFUNCTION_HELPER void ComputeHLLSpeeds_Roe_fwd(NewtonianIdealGasContext gas, State left, State dleft, CeedScalar u_left, CeedScalar du_left, 122 State right, State dright, CeedScalar u_right, CeedScalar du_right, CeedScalar *s_left, 123 CeedScalar *ds_left, CeedScalar *s_right, CeedScalar *ds_right) { 124 const CeedScalar gamma = HeatCapacityRatio(gas); 125 126 RoeWeights r = RoeSetup(left.U.density, right.U.density); 127 RoeWeights dr = RoeSetup_fwd(left.U.density, right.U.density, dleft.U.density, dright.U.density); 128 // Speed estimate 129 // Roe average eigenvalues for left and right non-linear waves. 130 // Stability requires that these speed estimates are *at least* as fast as the physical wave speeds. 131 CeedScalar u_roe = RoeAverage(r, u_left, u_right); 132 CeedScalar du_roe = RoeAverage_fwd(r, dr, u_left, u_right, du_left, du_right); 133 134 CeedScalar H_left = TotalSpecificEnthalpy(gas, left); 135 CeedScalar H_right = TotalSpecificEnthalpy(gas, right); 136 CeedScalar dH_left = TotalSpecificEnthalpy_fwd(gas, left, dleft); 137 CeedScalar dH_right = TotalSpecificEnthalpy_fwd(gas, right, dright); 138 139 CeedScalar H_roe = RoeAverage(r, H_left, H_right); 140 CeedScalar dH_roe = RoeAverage_fwd(r, dr, H_left, H_right, dH_left, dH_right); 141 CeedScalar a_roe = sqrt((gamma - 1) * (H_roe - 0.5 * Square(u_roe))); 142 CeedScalar da_roe = 0.5 * sqrt((gamma - 1) / (H_roe - 0.5 * Square(u_roe))) * dH_roe; // (da/dH) dH 143 da_roe -= 0.5 * sqrt(gamma - 1) * u_roe / sqrt(H_roe - 0.5 * Square(u_roe)) * du_roe; // (da/du) du 144 145 *s_left = u_roe - a_roe; 146 *ds_left = du_roe - da_roe; 147 *s_right = u_roe + a_roe; 148 *ds_right = du_roe + da_roe; 149 } 150 151 // ***************************************************************************** 152 // @brief Harten Lax VanLeer (HLL) approximate Riemann solver. 153 // Taking in two states (left, right) and returns RiemannFlux_HLL. 154 // The left and right states are specified from the perspective of an outward-facing normal vector pointing left to right. 155 // 156 // @param[in] gas NewtonianIdealGasContext for the fluid 157 // @param[in] left Fluid state of the domain interior (the current solution) 158 // @param[in] right Fluid state of the domain exterior (free stream conditions) 159 // @param[in] normal Normalized, outward facing boundary normal vector 160 // 161 // @return StateConservative with HLL Riemann Flux 162 // ***************************************************************************** 163 CEED_QFUNCTION_HELPER StateConservative RiemannFlux_HLL(NewtonianIdealGasContext gas, State left, State right, const CeedScalar normal[3]) { 164 StateConservative flux_left = FluxInviscidDotNormal(gas, left, normal); 165 StateConservative flux_right = FluxInviscidDotNormal(gas, right, normal); 166 167 CeedScalar u_left = Dot3(left.Y.velocity, normal); 168 CeedScalar u_right = Dot3(right.Y.velocity, normal); 169 170 CeedScalar s_left, s_right; 171 ComputeHLLSpeeds_Roe(gas, left, u_left, right, u_right, &s_left, &s_right); 172 173 // Compute HLL flux 174 if (0 <= s_left) { 175 return flux_left; 176 } else if (s_right <= 0) { 177 return flux_right; 178 } else { 179 return Flux_HLL(left, right, flux_left, flux_right, s_left, s_right); 180 } 181 } 182 183 // ***************************************************************************** 184 // @brief Forward-mode Derivative of Harten Lax VanLeer (HLL) approximate Riemann solver. 185 // 186 // @param gas NewtonianIdealGasContext for the fluid 187 // @param left Fluid state of the domain interior (the current solution) 188 // @param right Fluid state of the domain exterior (free stream conditions) 189 // @param dleft Derivative of fluid state of the domain interior (the current solution) 190 // @param dright Derivative of fluid state of the domain exterior (free stream conditions) 191 // @param normal Normalized, outward facing boundary normal vector 192 // 193 // @return StateConservative with derivative of HLL Riemann Flux 194 // ***************************************************************************** 195 CEED_QFUNCTION_HELPER StateConservative RiemannFlux_HLL_fwd(NewtonianIdealGasContext gas, State left, State dleft, State right, State dright, 196 const CeedScalar normal[3]) { 197 StateConservative flux_left = FluxInviscidDotNormal(gas, left, normal); 198 StateConservative flux_right = FluxInviscidDotNormal(gas, right, normal); 199 StateConservative dflux_left = FluxInviscidDotNormal_fwd(gas, left, dleft, normal); 200 StateConservative dflux_right = FluxInviscidDotNormal_fwd(gas, right, dright, normal); 201 202 CeedScalar u_left = Dot3(left.Y.velocity, normal); 203 CeedScalar u_right = Dot3(right.Y.velocity, normal); 204 CeedScalar du_left = Dot3(dleft.Y.velocity, normal); 205 CeedScalar du_right = Dot3(dright.Y.velocity, normal); 206 207 CeedScalar s_left, ds_left, s_right, ds_right; 208 ComputeHLLSpeeds_Roe_fwd(gas, left, dleft, u_left, du_left, right, dright, u_right, du_right, &s_left, &ds_left, &s_right, &ds_right); 209 210 if (0 <= s_left) { 211 return dflux_left; 212 } else if (s_right <= 0) { 213 return dflux_right; 214 } else { 215 return Flux_HLL_fwd(left, right, dleft, dright, flux_left, flux_right, dflux_left, dflux_right, s_left, s_right, ds_left, ds_right); 216 } 217 } 218 219 CEED_QFUNCTION_HELPER StateConservative RiemannFlux_HLLC_Star(NewtonianIdealGasContext gas, State side, StateConservative F_side, 220 const CeedScalar normal[3], CeedScalar u_side, CeedScalar s_side, CeedScalar s_star) { 221 CeedScalar fact = side.U.density * (s_side - u_side) / (s_side - s_star); 222 CeedScalar denom = side.U.density * (s_side - u_side); 223 // U_* = fact * star 224 StateConservative star = { 225 1.0, 226 { 227 side.Y.velocity[0] + (s_star - u_side) * normal[0], 228 side.Y.velocity[1] + (s_star - u_side) * normal[1], 229 side.Y.velocity[2] + (s_star - u_side) * normal[2], 230 }, 231 side.U.E_total / side.U.density // 232 + (s_star - u_side) * (s_star + side.Y.pressure / denom) 233 }; 234 return StateConservativeAXPBYPCZ(1, F_side, s_side * fact, star, -s_side, side.U); 235 } 236 237 CEED_QFUNCTION_HELPER StateConservative RiemannFlux_HLLC_Star_fwd(NewtonianIdealGasContext gas, State side, State dside, StateConservative F_side, 238 StateConservative dF_side, const CeedScalar normal[3], CeedScalar u_side, 239 CeedScalar du_side, CeedScalar s_side, CeedScalar ds_side, CeedScalar s_star, 240 CeedScalar ds_star) { 241 CeedScalar fact = side.U.density * (s_side - u_side) / (s_side - s_star); 242 CeedScalar dfact = (side.U.density * (ds_side - du_side) + dside.U.density * (s_side - u_side)) / (s_side - s_star) // 243 - fact / (s_side - s_star) * (ds_side - ds_star); 244 CeedScalar denom = side.U.density * (s_side - u_side); 245 CeedScalar ddenom = side.U.density * (ds_side - du_side) + dside.U.density * (s_side - u_side); 246 247 StateConservative star = { 248 1.0, 249 { 250 side.Y.velocity[0] + (s_star - u_side) * normal[0], 251 side.Y.velocity[1] + (s_star - u_side) * normal[1], 252 side.Y.velocity[2] + (s_star - u_side) * normal[2], 253 }, 254 side.U.E_total / side.U.density // 255 + (s_star - u_side) * (s_star + side.Y.pressure / denom) 256 }; 257 StateConservative dstar = { 258 0., 259 { 260 dside.Y.velocity[0] + (ds_star - du_side) * normal[0], 261 dside.Y.velocity[1] + (ds_star - du_side) * normal[1], 262 dside.Y.velocity[2] + (ds_star - du_side) * normal[2], 263 }, 264 dside.U.E_total / side.U.density - side.U.E_total / Square(side.U.density) * dside.U.density // 265 + (ds_star - du_side) * (s_star + side.Y.pressure / denom) // 266 + (s_star - u_side) * (ds_star + dside.Y.pressure / denom - side.Y.pressure / Square(denom) * ddenom) // 267 }; 268 269 const CeedScalar a[] = {1, ds_side * fact + s_side * dfact, s_side * fact, -ds_side, -s_side}; 270 const StateConservative U[] = {dF_side, star, dstar, side.U, dside.U}; 271 return StateConservativeMult(5, a, U); 272 } 273 274 // HLLC Riemann solver (from Toro's book) 275 CEED_QFUNCTION_HELPER StateConservative RiemannFlux_HLLC(NewtonianIdealGasContext gas, State left, State right, const CeedScalar normal[3]) { 276 StateConservative flux_left = FluxInviscidDotNormal(gas, left, normal); 277 StateConservative flux_right = FluxInviscidDotNormal(gas, right, normal); 278 279 CeedScalar u_left = Dot3(left.Y.velocity, normal); 280 CeedScalar u_right = Dot3(right.Y.velocity, normal); 281 CeedScalar s_left, s_right; 282 ComputeHLLSpeeds_Roe(gas, left, u_left, right, u_right, &s_left, &s_right); 283 284 // Contact wave speed; Toro (10.37) 285 CeedScalar rhou_left = left.U.density * u_left, rhou_right = right.U.density * u_right; 286 CeedScalar numer = right.Y.pressure - left.Y.pressure + rhou_left * (s_left - u_left) - rhou_right * (s_right - u_right); 287 CeedScalar denom = left.U.density * (s_left - u_left) - right.U.density * (s_right - u_right); 288 CeedScalar s_star = numer / denom; 289 290 // Compute HLLC flux 291 if (0 <= s_left) { 292 return flux_left; 293 } else if (0 <= s_star) { 294 return RiemannFlux_HLLC_Star(gas, left, flux_left, normal, u_left, s_left, s_star); 295 } else if (0 <= s_right) { 296 return RiemannFlux_HLLC_Star(gas, right, flux_right, normal, u_right, s_right, s_star); 297 } else { 298 return flux_right; 299 } 300 } 301 302 CEED_QFUNCTION_HELPER StateConservative RiemannFlux_HLLC_fwd(NewtonianIdealGasContext gas, State left, State dleft, State right, State dright, 303 const CeedScalar normal[3]) { 304 StateConservative flux_left = FluxInviscidDotNormal(gas, left, normal); 305 StateConservative flux_right = FluxInviscidDotNormal(gas, right, normal); 306 StateConservative dflux_left = FluxInviscidDotNormal_fwd(gas, left, dleft, normal); 307 StateConservative dflux_right = FluxInviscidDotNormal_fwd(gas, right, dright, normal); 308 309 CeedScalar u_left = Dot3(left.Y.velocity, normal); 310 CeedScalar u_right = Dot3(right.Y.velocity, normal); 311 CeedScalar du_left = Dot3(dleft.Y.velocity, normal); 312 CeedScalar du_right = Dot3(dright.Y.velocity, normal); 313 314 CeedScalar s_left, ds_left, s_right, ds_right; 315 ComputeHLLSpeeds_Roe_fwd(gas, left, dleft, u_left, du_left, right, dright, u_right, du_right, &s_left, &ds_left, &s_right, &ds_right); 316 317 // Contact wave speed; Toro (10.37) 318 CeedScalar rhou_left = left.U.density * u_left, drhou_left = left.U.density * du_left + dleft.U.density * u_left; 319 CeedScalar rhou_right = right.U.density * u_right, drhou_right = right.U.density * du_right + dright.U.density * u_right; 320 CeedScalar numer = right.Y.pressure - left.Y.pressure // 321 + rhou_left * (s_left - u_left) // 322 - rhou_right * (s_right - u_right); 323 CeedScalar dnumer = dright.Y.pressure - dleft.Y.pressure // 324 + rhou_left * (ds_left - du_left) + drhou_left * (s_left - u_left) // 325 - rhou_right * (ds_right - du_right) - drhou_right * (s_right - u_right); 326 CeedScalar denom = left.U.density * (s_left - u_left) - right.U.density * (s_right - u_right); 327 CeedScalar ddenom = left.U.density * (ds_left - du_left) + dleft.U.density * (s_left - u_left) // 328 - right.U.density * (ds_right - du_right) - dright.U.density * (s_right - u_right); 329 CeedScalar s_star = numer / denom; 330 CeedScalar ds_star = dnumer / denom - numer / Square(denom) * ddenom; 331 332 // Compute HLLC flux 333 if (0 <= s_left) { 334 return dflux_left; 335 } else if (0 <= s_star) { 336 return RiemannFlux_HLLC_Star_fwd(gas, left, dleft, flux_left, dflux_left, normal, u_left, du_left, s_left, ds_left, s_star, ds_star); 337 } else if (0 <= s_right) { 338 return RiemannFlux_HLLC_Star_fwd(gas, right, dright, flux_right, dflux_right, normal, u_right, du_right, s_right, ds_right, s_star, ds_star); 339 } else { 340 return dflux_right; 341 } 342 } 343