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ABSTRACT
Direct matrix operation is extremely memory-consuming to solve the basic equations of the radiation-hydrodynamics (R-HD) problems, e.g., Poisson equation and radiative transfer equation (RTE), especially in the cases of large grid number and multidimensions. In this work, the fast transform spectral method (FTSM), which requires much less memory than the direct matrix operation, is developed to avoid large dense matrix operation. The proposed method converges monotonically for arbitrary initial value and is verified via the multidimensional Poisson equations and one-dimensional RTE in cylindrical coordinate system. Benchmarks are also introduced to demonstrate the good accuracy of this method. The performance of the FTSM has been validated by comparing with the matrix multiplication transform spectral method (MMTSM). The results show that, the presented method has good robustness and accuracy, when the directly matrix operation of MMTSM is out of memory the FTSM still works well with high accuracy when the grid number is large enough. This means that the FTSM can be a better selection for the R-HD problems when large number grid is needed and especially for multidimensions.
Nomenclature
| a1 | = | degree of linear-anisotropy |
| a, c, j, k | = | index counters |
| c0 | = | speed of light, m/s |
| = | identified coefficients | |
| D1, D2, D3 | = | computational domain |
| = | boundaries | |
| er, eθ, ez | = | unit vectors |
| g1, g2 | = | boundary values |
| h1, h2 | = | initial values |
| i | = | complex unit, i2 = −1 |
| I | = | intensity of radiation, W/(m2 sr) |
| I0 | = | initial condition, W/(m2 sr) |
| N | = | grid number |
| nw | = | unit outward normal vector |
| P, Q | = | assembly variable |
| = | spectral coefficients | |
| q | = | radiative heat flux, W/m2 |
| T(r) | = | Chebyshev polynomial |
| = | time layers | |
| T | = | temperature, K |
| t | = | time, s |
| u, δ | = | normal variables |
| = | spectral coefficients | |
| = | quadrature weight | |
| α | = | interval |
| ατ, αf, αψ | = | standard coordinates |
| β | = | extinction coefficient, m−1 |
| ϵ | = | emissivity, relative error |
| θ, r, z | = | cylindrical coordinates |
| = | direction cosines | |
| ρ | = | source term |
| σ | = | Stefan–Boltzmann constant, 5.67 × 10−8 W/(m2 K4) |
| τ | = | optical coordinate, optical thickness, τ = βr |
| Φ | = | Scattering phase equation, sr−1 |
| = | polar and azimuthal angles, respectively, rad | |
| = | vector of radiation direction | |
| = | solid angle, sr | |
| ω | = | scattering albedo, m−1 |
| Subscripts | ||
| a, c, j, k | = | index counters |
| b | = | blackbody |
| benchmark | = | benchmark result |
| exact | = | exact solution |
| err | = | relative error |
| MAX | = | maximum value |
| N | = | index points |
| w | = | wall |
| = | directions |
| Superscripts | ||
| (l), (l − 1) | = | number of iterations |
| ′,″ | = | first and second orders |
| = | angular indexes |
Nomenclature
| a1 | = | degree of linear-anisotropy |
| a, c, j, k | = | index counters |
| c0 | = | speed of light, m/s |
| = | identified coefficients | |
| D1, D2, D3 | = | computational domain |
| = | boundaries | |
| er, eθ, ez | = | unit vectors |
| g1, g2 | = | boundary values |
| h1, h2 | = | initial values |
| i | = | complex unit, i2 = −1 |
| I | = | intensity of radiation, W/(m2 sr) |
| I0 | = | initial condition, W/(m2 sr) |
| N | = | grid number |
| nw | = | unit outward normal vector |
| P, Q | = | assembly variable |
| = | spectral coefficients | |
| q | = | radiative heat flux, W/m2 |
| T(r) | = | Chebyshev polynomial |
| = | time layers | |
| T | = | temperature, K |
| t | = | time, s |
| u, δ | = | normal variables |
| = | spectral coefficients | |
| = | quadrature weight | |
| α | = | interval |
| ατ, αf, αψ | = | standard coordinates |
| β | = | extinction coefficient, m−1 |
| ϵ | = | emissivity, relative error |
| θ, r, z | = | cylindrical coordinates |
| = | direction cosines | |
| ρ | = | source term |
| σ | = | Stefan–Boltzmann constant, 5.67 × 10−8 W/(m2 K4) |
| τ | = | optical coordinate, optical thickness, τ = βr |
| Φ | = | Scattering phase equation, sr−1 |
| = | polar and azimuthal angles, respectively, rad | |
| = | vector of radiation direction | |
| = | solid angle, sr | |
| ω | = | scattering albedo, m−1 |
| Subscripts | ||
| a, c, j, k | = | index counters |
| b | = | blackbody |
| benchmark | = | benchmark result |
| exact | = | exact solution |
| err | = | relative error |
| MAX | = | maximum value |
| N | = | index points |
| w | = | wall |
| = | directions |
| Superscripts | ||
| (l), (l − 1) | = | number of iterations |
| ′,″ | = | first and second orders |
| = | angular indexes |