*Gary A. Glatzmaier*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter modifies the numerical code by adding the nonlinear terms to produce finite-amplitude simulations. The nonlinear terms are calculated using a Galerkin method in spectral space. After ...
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This chapter modifies the numerical code by adding the nonlinear terms to produce finite-amplitude simulations. The nonlinear terms are calculated using a Galerkin method in spectral space. After explaining the modifications to the linear model, the chapter shows how to add the nonlinear terms to the code. It also discusses the Galerkin method, the strategy of computing the contribution to the nonlinear terms for each mode due to the binary interactions of many other modes. The Galerkin method works fine as far as calculating the nonlinear terms is concerned because of the simple geometry and convenient boundary conditions. The chapter concludes by showing how to construct a nonlinear code and performing nonlinear simulations.Less

This chapter modifies the numerical code by adding the nonlinear terms to produce finite-amplitude simulations. The nonlinear terms are calculated using a Galerkin method in spectral space. After explaining the modifications to the linear model, the chapter shows how to add the nonlinear terms to the code. It also discusses the Galerkin method, the strategy of computing the contribution to the nonlinear terms for each mode due to the binary interactions of many other modes. The Galerkin method works fine as far as calculating the nonlinear terms is concerned because of the simple geometry and convenient boundary conditions. The chapter concludes by showing how to construct a nonlinear code and performing nonlinear simulations.

*Gary A. Glatzmaier*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0008
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter focuses on time integration schemes, including fourth-order accurate Runge–Kutta and predictor-corrector schemes as well as schemes that allow larger time steps (and therefore fewer ...
More

This chapter focuses on time integration schemes, including fourth-order accurate Runge–Kutta and predictor-corrector schemes as well as schemes that allow larger time steps (and therefore fewer steps for a given amount of simulated time) by treating the linear diffusion terms implicitly. The nonlinear terms, however, couple all the modes and so would be extremely expensive to treat implicitly; therefore they are usually treated explicitly. Such “semi-implicit” schemes considerably improve the efficiency of the computer code. The chapter also describes the Crank–Nicolson scheme and concludes by showing how the current numerical model can easily be modified to study mantle convection (also called “geodynamics”) using the vorticity equation in the limit of an infinite Prandtl number.Less

This chapter focuses on time integration schemes, including fourth-order accurate Runge–Kutta and predictor-corrector schemes as well as schemes that allow larger time steps (and therefore fewer steps for a given amount of simulated time) by treating the linear diffusion terms implicitly. The nonlinear terms, however, couple all the modes and so would be extremely expensive to treat implicitly; therefore they are usually treated explicitly. Such “semi-implicit” schemes considerably improve the efficiency of the computer code. The chapter also describes the Crank–Nicolson scheme and concludes by showing how the current numerical model can easily be modified to study mantle convection (also called “geodynamics”) using the vorticity equation in the limit of an infinite Prandtl number.

*George Em Karniadakis and Spencer J. Sherwin*

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198528692
- eISBN:
- 9780191713491
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528692.003.0008
- Subject:
- Mathematics, Numerical Analysis

This chapter presents different ways of formulating the incompressible Navier-Stokes equations based on primitive variables, that is, velocity and pressure, as well as velocity-vorticity algorithms. ...
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This chapter presents different ways of formulating the incompressible Navier-Stokes equations based on primitive variables, that is, velocity and pressure, as well as velocity-vorticity algorithms. It considers both coupled, splitting, and least-squares formulations for primitive variables. Both the Uzawa coupled algorithm and a new substructured solver are discussed. The discussion on primitive variables time-splitting includes recent theoretical advances in the pressure-correction and velocity correction schemes as well as the rotational formulation of the pressure boundary condition. The final section is devoted to nonlinear terms; it includes a discussion of spatial and temporal discretization with focus on the semi-Lagrangian method for the incompressible Navier-Stokes equations.Less

This chapter presents different ways of formulating the incompressible Navier-Stokes equations based on primitive variables, that is, velocity and pressure, as well as velocity-vorticity algorithms. It considers both coupled, splitting, and least-squares formulations for primitive variables. Both the Uzawa coupled algorithm and a new substructured solver are discussed. The discussion on primitive variables time-splitting includes recent theoretical advances in the pressure-correction and velocity correction schemes as well as the rotational formulation of the pressure boundary condition. The final section is devoted to nonlinear terms; it includes a discussion of spatial and temporal discretization with focus on the semi-Lagrangian method for the incompressible Navier-Stokes equations.