|
288 | 288 | }, |
289 | 289 | "outputs": [], |
290 | 290 | "source": [ |
291 | | - "# we use the helper rountines above to find the orthogonal left and right eigenvectors\n", |
| 291 | + "# we use the helper routines above to find the orthogonal left and right eigenvectors\n", |
292 | 292 | "eigen = eigensystem(A)" |
293 | 293 | ] |
294 | 294 | }, |
|
488 | 488 | "source": [ |
489 | 489 | "## $\\beta$'s and final update\n", |
490 | 490 | "\n", |
491 | | - "The final interface state is writen by projecting the jump in primitive variables, $\\Delta q$, into characteristic variables (as $l \\cdot \\Delta q$), and then adding up all the jumps that reach the interface.\n", |
| 491 | + "The final interface state is written by projecting the jump in primitive variables, $\\Delta q$, into characteristic variables (as $l \\cdot \\Delta q$), and then adding up all the jumps that reach the interface.\n", |
492 | 492 | "\n", |
493 | 493 | "The convention is to write $\\beta^\\nu = l^\\nu \\cdot \\Delta q$, where the superscript identifies which eigenvalue (and corresponding eigenvectors) we are considering. Note, that often a reference state is used, and the jump, $\\Delta q$, will be the difference with respect to this reference state. For PPM, the $\\Delta q$ will take the form of the integral under the parabola over the range that each wave can reach.\n", |
494 | 494 | "\n", |
|
877 | 877 | }, |
878 | 878 | "outputs": [], |
879 | 879 | "source": [ |
880 | | - "# we use the helper rountines above to find the orthogonal left and right eigenvectors\n", |
| 880 | + "# we use the helper routines above to find the orthogonal left and right eigenvectors\n", |
881 | 881 | "eigen = eigensystem(A)" |
882 | 882 | ] |
883 | 883 | }, |
|
1544 | 1544 | }, |
1545 | 1545 | "outputs": [], |
1546 | 1546 | "source": [ |
1547 | | - "# we use the helper rountines above to find the orthogonal left and right eigenvectors\n", |
| 1547 | + "# we use the helper routines above to find the orthogonal left and right eigenvectors\n", |
1548 | 1548 | "eigen = eigensystem(A, suba=cg, subb=sqrt(cc))" |
1549 | 1549 | ] |
1550 | 1550 | }, |
|
2092 | 2092 | "source": [ |
2093 | 2093 | "# Gray FLD Radiation Euler Equations with $(\\gamma_e)$\n", |
2094 | 2094 | "\n", |
2095 | | - "We now look at the same system with a different auxillary thermodynamic variable (as we did with pure hydro), using $q = (\\tau, u, p, {\\gamma_e}_g, E_r)^\\intercal$:\n", |
| 2095 | + "We now look at the same system with a different auxiliary thermodynamic variable (as we did with pure hydro), using $q = (\\tau, u, p, {\\gamma_e}_g, E_r)^\\intercal$:\n", |
2096 | 2096 | "\\begin{align}\n", |
2097 | 2097 | "\\frac{\\partial \\tau}{\\partial t} &= -u\\frac{\\partial \\tau}{\\partial x} + \\tau \\frac{\\partial u}{\\partial x} \\\\\n", |
2098 | 2098 | "\\frac{\\partial u}{\\partial t} &= -u \\frac{\\partial u}{\\partial x} - \\tau \\frac{\\partial p}{\\partial x}\n", |
|
2295 | 2295 | }, |
2296 | 2296 | "outputs": [], |
2297 | 2297 | "source": [ |
2298 | | - "# we use the helper rountines above to find the orthogonal left and right eigenvectors\n", |
| 2298 | + "# we use the helper routines above to find the orthogonal left and right eigenvectors\n", |
2299 | 2299 | "eigen = eigensystem(A, suba=cg, subb=sqrt(cc))" |
2300 | 2300 | ] |
2301 | 2301 | }, |
|
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