Abstract
In this work, we construct novel discretizations for the unsteady convection–diffusion equation. Our discretization relies on multiderivative time integrators together with a novel discretization that reduces the total number of unkowns for the solver. These type of temporal discretizations come from an umbrella class of methods that include Lax–Wendroff (Taylor) as well as Runge–Kutta methods as special cases. We include two-point collocation methods with multiple time derivatives as well as a sixth-order fully implicit collocation method that only requires a total of three stages. Numerical results for a number of sample linear problems indicate the expected order of accuracy and indicate we can take arbitrarily large time steps.
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Al-Rabeh, A.H.: Embedded DIRK methods for the numerical integration of stiff systems of ODEs. Int. J. Comput. Math. 21, 65–84 (1987)
Alexander, R.: Diagonally implicit Runge–Kutta methods for stiff O.D.E’.s. SIAM J. Numer. Anal. 14, 1006–1021 (1977)
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)
Balay, S.,Brown, J., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc users manual. Technical report ANL-95/11 - Revision 3.1, Argonne National Laboratory (2010)
Balay, S., Brown, J., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc web page (2011). http://www.mcs.anl.gov/petsc
Balay, S., Gropp, W.D., McInnes, L.C., Smith, B.F.: Efficient management of parallelism in object oriented numerical software libraries. In: Arge, E., Bruaset, A.M., Langtangen, H.P. (eds.) Modern Software Tools in Scientific Computing, pp. 163–202. Birkhäuser Press, Boston (1997)
Balsara, D.S., Kim, J.: A subluminal relativistic magnetohydrodynamics scheme with ADER-WENO predictor and multidimensional Riemann solver-based corrector. J. Comput. Phys. 312, 357–384 (2016)
Butcher, J.: General linear methods. Acta Numer. 15, 157–256 (2006)
Butcher, J.C.: Implicit Runge–Kutta processes. Math. Comput. 18, 50–64 (1964)
Butcher, J.C.: On the convergence of numerical solutions to ordinary differential equations. Math. Comput. 20(93), 1–10 (1966)
Cash, J.: Diagonally implicit Runge–Kutta formulae with error estimates. J. Inst. Math. Appl. 24, 293–301 (1979)
Christlieb, A.J., Feng, X., Seal, D.C., Tang, Q.: A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations. J. Computat. Phys. 316, 218–242 (2016)
Christlieb, A.J., Gottlieb, S., Grant, Z., Seal, D.C.: Explicit strong stability preserving multistage two-derivative time-stepping schemes. J. Sci. Comput. 68(3), 914–942 (2016)
Christlieb, A.J., Güçlü, Y., Seal, D.C.: The Picard integral formulation of weighted essentially nonoscillatory schemes. SIAM J. Numer. Anal. 53(4), 1833–1856 (2015)
Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. J. Comput. Phys. 314, 824–862 (2016)
Ehle, B.L.: High order \(A\)-stable methods for the numerical solution of systems of D.E’.s. BIT Numer. Math. 8, 276–278 (1968)
Ehle, B.L.: \(A\)-stable methods and Padé approximations to the exponential. SIAM J. Math. Anal. 4, 671–680 (1973)
Enright, W.H.: Second derivative multistep methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 11, 321–331 (1974)
Gekeler, E., Widmann, R.: On the order conditions of Runge–Kutta methods with higher derivatives. Numer. Math. 50, 183–203 (1986)
Gekeler, E.W.: On implicit Runge–Kutta methods with higher derivatives. BIT Numer. Math. 28, 809–816 (1988)
Genin, Y.: An algebraic approach to \(A\)-stable linear multistep-multiderivative integration formulas. BIT Numer. Math. 14, 382–406 (1974)
Guo, W., Qiu, J.-M., Qiu, J.: A new Lax–Wendroff discontinuous Galerkin method with superconvergence. J. Sci. Comput. 65(1), 299–326 (2015)
Hairer, E., Wanner, G.: Multistep-multistage-multiderivative methods for ordinary differential equations. Computing 11(3), 287–303 (1973)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer Series in Computational Mathematics. Springer, Berlin (1991)
Harten, A., Enquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys. 71, 231–303 (1987)
Houston, P., Schwab, C., Süli, E.: Discontinuous \(hp\)-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39, 2133–2163 (2002)
Houston, P., Süli, E.: hp-adaptive discontinuous Galerkin finite element methods for first order hyperbolic problems. SIAM J. Sci. Comput. 23, 1226–1252 (2001)
Jaust, A., Schütz, J., Seal, D.C.: Implicit multistage two-derivative discontinuous Galerkin schemes for viscous conservation laws. J. Sci. Comput. 69, 866–891 (2016)
Jeltsch, R.: A necessary condition for \(A\)-stability of multistep multiderivative methods. Math. Comput. 30(136), 739–746 (1976)
Jiang, Y., Shu, C.-W., Zhang, M.: An alternative formulation of finite difference weighted ENO schemes with Lax–Wendroff time discretization for conservation laws. SIAM J. Sci. Comput. 35(2), A1137–A1160 (2013)
Kastlunger, K., Wanner, G.: On Turan type implicit Runge–Kutta methods. Computing 9, 317–325 (1972)
Lax, P., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13(2), 217–237 (1960)
Li, J., Du, Z.: A two-stage fourth order time-accurate discretization for Lax–Wendroff type flow solvers I. Hyperbolic conservation laws. SIAM J. Sci. Comput. 38(5), A3046–A3069 (2016)
Meir, A., Sharma, A.: An extension of Obreshkov’s formula. SIAM J. Numer. Anal. 5, 488–490 (1968)
Moe, S.A., Rossmanith, J.A., Seal, D.C.: Positivity-preserving discontinuous Galerkin methods with Lax–Wendroff time discretizations. J. Sci. Comput. 71(1), 44–70 (2017)
Mühlbach, G.: An algorithmic approach to Hermite–Birkhoff-interpolation. Numer. Math. 37, 339–347 (1981)
Pan, L. Li, J., Xu, K.: A few benchmark test cases for higher-order Euler solvers (2016). arXiv:1609.04491
Pan, L., Xu, K., Li, Q., Li, J.: An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier–Stokes equations. J. Comput. Phys. 326, 197–221 (2016)
Qiu, J., Dumbser, M., Shu, C.-W.: The discontinuous Galerkin method with Lax–Wendroff type time discretizations. Comput. Methods Appl. Mech. Eng. 194(42–44), 4528–4543 (2005)
Qiu, J., Shu, C.-W.: Finite difference WENO schemes with Lax–Wendroff-type time discretizations. SIAM J. Sci. Comput. 24(6), 2185–2198 (2003)
Rappaport, K.D.: S. Kovalevsky: a mathematical lesson. Am. Math. Mon. 88(8), 564–574 (1981)
Rosen, J. S.: Numerical solution of differential equations using Obrechkoff corrector formulas. Technical Note M527, Institute for Computer Applications in Science and Engineering, George C. Marshall Space Flight Center, Marshall, AL (1969)
Schöberl, J.: Netgen—an advancing front 2d/3d-mesh generator based on abstract rules. Comput. Vis. Sci. 1, 41–52 (1997)
Schoenberg, I.J.: On Hermite-Birkhoff interpolation. J. Math. Anal. Appl. 16, 538–543 (1966)
Seal, D.C., Güçlü, Y., Christlieb, A.: High-order multiderivative time integrators for hyperbolic conservation laws. J. Sci. Comput. 60, 101–140 (2014)
Seal, D.C., Tang, Q., Xu, Z., Christlieb, A.J.: An explicit high-order single-stage single-step positivity-preserving finite difference WENO method for the compressible Euler equations. J. Sci. Comput. 68(1), 171–190 (2016)
Stroud, A.H., Stancu, D.D.: Quadrature formulas with multiple Gaussian nodes. SIAM J. Numer. Anal. 2, 129–143 (1965)
Toro, E.F., Titarev, V. A.: Solution of the generalized Riemann problem for advection-reaction equations. In: Proceedings: Mathematical, Physical and Engineering Sciences, vol. 458, no. 2018, pp. 271–281 (2002)
Tsai, A., Chan, R., Wang, S.: Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach. Numer. Algorithms 65, 687–703 (2014)
Vos, P.E., Eskilsson, C., Bolis, A., Chun, S., Kirby, R.M., Sherwin, S.J.: A generic framework for time-stepping partial differential equations (PDEs): general linear methods, object-oriented implementation and application to fluid problems. Int. J. Comput. Fluid Dyn. 25(3), 107–125 (2011)
Yakubu, D., Kwami, A.: Implicit two-derivative Runge–Kutta collocation methods for systems of initial value problems. J. Niger. Math, Soc. 34, 128–142 (2015)
Zanotti, O., Fambri, F., Dumbser, M., Hidalgo, A.: Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Comput. Fluids 118, 204–224 (2015)
Zorío, D., Baeza, A., Mulet, P.: An approximate Lax–Wendroff-type procedure for high order accurate schemes for hyperbolic conservation laws. J. Sci. Comput. 71, 246–273 (2017)
Acknowledgements
D. Seal acknowledges funding by the Naval Academy Research Council. The study of A. Jaust was supported by the Special Research Fund (BOF) of Hasselt University (Grant No. BOF16DOC02).
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Schütz, J., Seal, D.C. & Jaust, A. Implicit Multiderivative Collocation Solvers for Linear Partial Differential Equations with Discontinuous Galerkin Spatial Discretizations. J Sci Comput 73, 1145–1163 (2017). https://doi.org/10.1007/s10915-017-0485-9
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DOI: https://doi.org/10.1007/s10915-017-0485-9