Treatment of the polar coordinate singularity in axisymmetric wave propagation using high-order summation-by-parts operators on a staggered grid. (English) Zbl 1390.76613
Summary: We develop a high-order finite difference scheme for axisymmetric wave propagation in a cylindrical conduit filled with a viscous fluid. The scheme is provably stable, and overcomes the difficulty of the polar coordinate singularity in the radial component of the diffusion operator. The finite difference approximation satisfies the principle of summation-by-parts (SBP), which is used to establish stability using the energy method. To treat the coordinate singularity without losing the SBP property of the scheme, a staggered grid is introduced and quadrature rules with weights set to zero at the endpoints are considered. The accuracy of the scheme is studied both for a model problem with periodic boundary conditions at the ends of the conduit and its practical utility is demonstrated by modeling acoustic-gravity waves in a magmatic conduit.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
polar coordinate singularity; summation-by-parts; high-order finite difference method; staggered grid; axisymmetric wave propagationReferences:
| [1] | Womersley, J. R., Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J Physiol, 127, 3, 553-563, (1955) |
| [2] | Atabek, H. B.; Lew, H. S., Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube, Biophys J, 6, 4, 481-503, (1966) |
| [3] | Karlstrom, L.; Dunham, E. M., Excitation and resonance of acoustic-gravity waves in a column of stratified, bubbly magma, J Fluid Mech, 797, 431-470, (2016) · Zbl 1422.86016 |
| [4] | Tavakoli, E.; Lessani, B.; Hosseini, R., High-order pole-treatment in cylindrical coordinates for incompressible flow simulations with finite-difference collocated schemes, J Comput Phys, 296, 1-24, (2015) · Zbl 1352.76085 |
| [5] | Oguic, R.; Viazzo, S.; Poncet, S., A parallelized multidomain compact solver for incompressible turbulent flows in cylindrical geometries, J Comput Phys, 300, 710-731, (2015) · Zbl 1349.76515 |
| [6] | Griffin, M. D.; Jones, E.; Anderson, J. D., A computational fluid dynamic technique valid at the centerline for non-axisymmetric problems in cylindrical coordinates, J Comput Phys, 30, 3, 352-360, (1979) · Zbl 0399.76063 |
| [7] | Constantinescu, G.; Lele, S., A highly accurate technique for the treatment of flow equations at the polar axis in cylindrical coordinates using series expansions, J Comput Phys, 183, 1, 165-186, (2002) · Zbl 1058.76580 |
| [8] | Morinishi, Y.; Vasilyev, O. V.; Ogi, T., Fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations, J Comput Phys, 197, 2, 686-710, (2004) · Zbl 1079.76602 |
| [9] | Mohseni, K.; Colonius, T., Numerical treatment of polar coordinate singularities, J Comput Phys, 157, 2, 787-795, (2000) · Zbl 0981.76075 |
| [10] | Verzicco, R.; Orlandi, P., A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates, J Comput Phys, 123, 2, 402-414, (1996) · Zbl 0849.76055 |
| [11] | Kreiss, H.-O.; Scherer, G., Finite element and finite difference methods for hyperbolic partial differential equations, (Boor, C.d., Mathematical Aspects of Finite Elements in Partial Differential Equations, (1974), Academic Press), 195-212 · Zbl 0355.65085 |
| [12] | Strand, B., Summation by parts for finite difference approximations for d/dx, J Comput Phys, 110, 1, 47-67, (1994) · Zbl 0792.65011 |
| [13] | Svärd, M.; Nordström, J., Review of summation-by-parts schemes for initial-boundary-value problems, J Comput Phys, 268, 17-38, (2014) · Zbl 1349.65336 |
| [14] | Carpenter, M. H.; Gottlieb, D.; Abarbanel, S., Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, J Comput Phys, 111, 2, 220-236, (1994) · Zbl 0832.65098 |
| [15] | Fernández, D. C.D. R.; Hicken, J. E.; Zingg, D. W., Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations, Comput Fluids, 95, 171-196, (2014) · Zbl 1390.65064 |
| [16] | Calabrese, G.; Neilsen, D., Spherical excision for moving black holes and summation by parts for axisymmetric systems, Phys Rev D, 69, 044020, (2004) |
| [17] | Gundlach, C.; Martín-García, J. M.; Garfinkle, D., Summation by parts methods for spherical harmonic decompositions of the wave equation in any dimensions, Classical Quantum Gravity, 30, 14, 145003, (2013) · Zbl 1273.83056 |
| [18] | Carpenter, M. H.; Kennedy, C. A., Fourth-order 2n-storage Runge-Kutta schemes, Tech. Rep. NASA-TM-109112, (1994), NASA |
| [19] | Gustafsson, B., The convergence rate for difference approximations to mixed initial boundary value problems, Math Comput, 29, 130, 396-406, (1975) · Zbl 0313.65085 |
| [20] | Svärd, M.; Nordström, J., On the order of accuracy for difference approximations of initial-boundary value problems, J Comput Phys, 218, 1, 333-352, (2006) · Zbl 1123.65088 |
| [21] | Larsson, M.; Müller, B., Numerical simulation of confined pulsating jets in human phonation, Comput Fluids, 38, 7, 1375-1383, (2009) · Zbl 1242.76374 |
| [22] | Nilsson, S.; Petersson, N. A.; Sjögreen, B.; Kreiss, H.-O., Stable difference approximations for the elastic wave equation in second order formulation, SIAM J Numer Anal, 45, 5, 1902-1936, (2007) · Zbl 1158.65064 |
| [23] | Mattsson, K., Summation by parts operators for finite difference approximations of second-derivatives with variable coefficients, J Sci Comput, 51, 3, 650-682, (2012) · Zbl 1252.65055 |
| [24] | Sjögreen, B.; Petersson, N. A., A fourth order accurate finite difference scheme for the elastic wave equation in second order formulation, J Sci Comput, 52, 1, 17-48, (2012) · Zbl 1255.65162 |
| [25] | Fernández, D. C.D. R.; Zingg, D. W., Generalized summation-by-parts operators for the second derivative, SIAM Journal on Scientific Computing, 37, 6, A2840-A2864, (2015) · Zbl 1329.65253 |
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