Questions tagged [pde]
Partial differential equations (PDEs) are equations that relate the partial derivatives of a function of more than one variable. This tag is intended for questions on modeling phenomena with PDEs, solving PDEs, and other related aspects.
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Are there operator splitting approaches for multiphysics PDEs that achieve high order convergence?
Given an evolution PDE
$$u_t = Au + Bu$$
where $A,B$ are (possibly nonlinear) differential operators that don't commute, a common numerical approach is to alternate between solving
$$u_t = Au$$
...
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14
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1
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Is there a multigrid algorithm that solves Neumann problems and has a convergence rate independent of the number of levels?
Multigrid methods usually solve Dirichlet problems on levels (e.g. point Jacobi or Gauss-Seidel). When using continuous finite element methods, it is much less expensive to assemble small Neumann ...
- 25.7k
13
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How to construct well-balanced finite volume and discontinuous Galerkin methods for hyperbolic PDEs with source terms?
Source terms, such as those due to bathymetry in the shallow water equations, need to be integrated in a special way in order to preserve physical steady states. Is there a general way to construct ...
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4
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Which methods can ensure that physical quantities remain positive throughout a PDE simulation?
Physical quantities like pressure, density, energy, temperature, and concentration should always be positive, but numerical methods sometimes compute negative values during the solution process. This ...
- 25.7k
14
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What spatial discretizations work for incompressible flow with anisotropic boundary meshes?
High Reynolds number flows produce very thin boundary layers. If wall resolution is used in Large Eddy Simulation, the aspect ratio may be on the order of $10^6$. Many methods become unstable in this ...
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4
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Why is local conservation important when solving PDEs?
Engineers often insist on using locally conservative methods such as finite volume, conservative finite difference, or discontinuous Galerkin methods for solving PDEs.
What can go wrong when using a ...
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15
votes
3
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What is a scalable preconditioner for high-frequency Helmholtz?
Standard multigrid and domain decomposition methods do not work, but I have large 3D problems and direct solvers are not an option. What methods should I try?
How are my choices affected by the ...
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When should implicit methods be used in the integration of hyperbolic PDEs?
Numerical methods for solving PDEs (or ODEs) fall into two broad categories: explicit and implicit methods. Implicit methods allow larger stable timesteps but require more work per step. For ...
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2
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How can I choose a good Riemann solver when numerically solving a system of hyperbolic PDEs?
Many numerical methods for hyperbolic PDEs are based on the use of Riemann solvers. Such solvers are essential for accurately capturing shock waves. There are a range of such solvers available for ...
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Why is Newton's method not converging?
I am using PETSc's nonlinear solver package SNES to solve a system of nonlinear equations obtained by discretizing a partial differential equation. How can I determine why the solver is not converging ...
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