Questions tagged [finite-difference]

Referring to the discretization of derivatives by Finite differences, and its applications to numerical solutions of partial differential equations.

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Implementation of boundary conditions for 1D Euler equations

I'm trying to solve 1D Euler equations with gravity in spherical coordinates using a finite-difference TVD MacCormack method on a non-uniform grid of $N$ components, following the method provided in ...
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Verification on pressure predictor method for CFD code

I have developed a python code for a lid-drive cavity model. However, my results are not converging. The algorithm of my code looks like this: Euler Momentum Equation looks like this: $$\frac{u^{n+1}...
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FDM discretization of equation on the boundary

In order to simulate the following equation using FDM $$u_t(t,x)-u_{xx}(t,x)=0, \quad (t,x) \in (0,1)\times (0,1)$$ $$(u_t(t,x)-u_{x}(t,x))\rvert_{x=0}=0, \quad t \in (0,1)$$ $$(u_t(t,x)+u_{x}(t,x))\...
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WENO scheme on curvilinear coordinates

I've been developing a curvilinear FVM code. So far I've implemented the PPM scheme and am looking into adding WENO schemes. So far I've been discretizing the grid metrics using a second-order central....
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Need an example Legendre-Gauss-Radau pseudospectral differentiation matrix or Matlab code

I'm trying to implement various kinds of pseudospectral methods for direct optimization in Matlab using IPOPT. I've got some working Legendre-Gauss-Lobatto code, but would like to use the flipped ...
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1answer
93 views

Solving nonlinear PDE with finite difference based on Newton-Krylov

I am now working on solving MHD equations with finite difference method, which include nonlinear equations: $$ \frac{\partial\rho}{\partial t}+\nabla\cdot\left[\left(\rho_0+\rho\right){v}\right]-\...
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87 views

What method of Finite difference is this?

I am reviewing Numerical Recipes method on solving ODEs via relaxation (Chapter 18.3 in the 3rd edition) and they chose a finite difference method I am unfamiliar with (Equation 18.3.2): \begin{...
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38 views

Interpolating the gradient of a cylindrically symmetric potential field that's 'supposed to' obey the Laplace equation?

The script below tries to implement a Jacobi iterative relaxation of a potential field for an electrostatic lens. It's hot-off-the-press and I've just started to debug and look for things to test it ...
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Dealing with spurious oscillations in particle tracking methods

I work on modelling high intensity discharge xenon-filled lamps. The model governing the discharge is quite complex and sadly includes fluid dynamics. After some time, I managed to implement a finite-...
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1answer
78 views

Finite difference methods

I am currently applying the finite difference method to the solution of the diffusion equation. I think that a problem has occurred, and is as follows, my explicit method is the most accurate when ...
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35 views

Neumann boundary conditions on arbitrary surface for finite difference diffusion

I am facing the following problem, formulated in practical terms: I have a region $\Omega$ in two or three dimensions, represented as a binary mask, and an initial density $u_0$ within that region ...
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Second derivative using Fornberg finite difference method

I have some discrete data, non-equispaced in x, y=f(x). I want to use a numerical finite difference method to calculate the second derivatives of y, at some point. I am using the Fornberg method, ...
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How to determine the order of convergence of the Euler-Maruyama method?

This question is originally posted in Quant.StackExchange but has been unanswered for some time so I ask in here. To make this simple let us consider the Geometric Brownian Motions (GBM). My ...
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Finite difference for a highly nonlinear equation - The wind within the forest

Based on the Navier-Stokes equations and a few parameterizations, the horizontal steady-state wind $u(z)$ within a forest of height H satisfies: $$ a\left(\frac{du}{dz}\right)^2 + b\frac{du}{dz} \...
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61 views

Dirichlet boundary conditions in the 1D Heat Equation

Please consider the assignment I have uploaded on the picture. I am confused about the functions $g_L(t)$, $g_R(t)$ and $\eta(x)$. What are they and how do I find them... My question: Is it possbile ...
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Damped Harmonic Oscillation. Efficient algorithm to find the parameters resulting in threshold oscillation amplitude

Let's assume, that we have damped harmonic oscillation of a body in the form of a cone, immersed in a liquid. Equilibrium condition of the body is: $$m\overrightarrow{a} = \overrightarrow{F_\text{...
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1answer
111 views

Numerically solving a non-linear PDE

I have this non-linear partial differential equation. $$ \frac{\partial C}{\partial t}=\left(\frac{\partial C}{\partial x}\right)^2+C\frac{\partial^2 C}{\partial x^2} $$ I want to use the finite ...
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Double mach reflection at a inclined wedge

I am running into a strange problem when solving the 2D compressible Euler equations on a inclined wedge. To elaborate, my top boundary condition seems to emitting some type of instability. I have ...
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59 views

Numerical Stability of a Generalized Spatial Discretization Scheme

After reading the matrix stability chapter (10) of Hirsch [1], I decided to dive in the reference list of the chapter. One of the papers [2], which is cited as reference shows an very interesting ...
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Computationally obtaining the convergence rate of upwind scheme for Advection equation

The Advection equation (with velocity = 1) is $${\partial u \over \partial x} + {\partial u \over \partial t} = 0$$ I am trying to solve the equation with periodic BC. One of the ways to numerically ...
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Stability of Crank-Nicolson for $u_t = iu_{xx}+2iu$

I want to use the Crank-Nicolson scheme to solve the equation $$u_t = iu_{xx}+2iu$$ Here's the analysis: Suppose we make a grid, with $k = dt$ and $h = dx$, the usual notation, and also $u_j^n = u(...
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Boundary conditions for a Non-linear Schrödinger equation using an extended crank nicolson scheme

I try to solve numerically the following PDE for $E(r, z)$ with a cylindrical symmetrie (i. e. $E(r, z) = E(-r, z)$). $\frac{\partial E}{\partial z} = \frac{i}{2k} \Delta E + \mathcal{N}(E)$ Where $...
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Numerical methods for the continuity equation with Sobolev vector field

Consider the continuity equation $$ \partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N, $$ with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
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104 views

How to obtain linear tridiagonal system from PDE

I'm trying to re-solve the governing equations in hydraulic fracturing modeling as instructed step by step in a paper. After (A-9), the author stated that by substituting A-6, A-8 and A-9 into ...
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70 views

CFL equation for non-linear equation

I am trying to solve numerically (obviously) inviscid Burgers' equation with the finite difference method. The equation is the following: $$ \displaystyle \partial_t u + u \, \partial_x u = 0 $$ ...
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74 views

Good C, C++ library for efficient grid search / tuples, ideally with bindings to Eigen

I have a $q$-dimensional grid, known at run, not compile-time, that has $50$ points in each direction and hence $50^3$ combinations that I would like to first build and then call a function with each ...
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Use of non-typical values of $\theta$ in theta-methods

The theta-method is a popular solution for solving time-transient PDEs (or ODEs), which consists of solving the general equation for each time step: $$ \frac{u^{n+1} - u^{n}}{\Delta t} + (\theta f(u^{...
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High-accuracy numerical differentiation

I have a $200 \times 200$ matrix representing the values taken by a function over an equally spaced grid. I would like to perform derivatives on it. I am interested in its gradient (i.e. its ...
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1answer
66 views

Error for the finite differences scheme — Advection equation

Consider the advection equation (1D in space) $$ \frac{\partial u}{\partial t} + V\, \frac{\partial u}{\partial x}=0 $$ and we solve it numerically on $[0,1]\times [0,1]\ni (t,x)$ using a forward ...
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1answer
67 views

Approximation Error in a Finite Difference Approximation of the Square of Derivative

First Part: (First-order derivative) Assuming $f$ is an infinitely differential function everywhere, the Taylor series of $f(x + h)$ at $x$ is \begin{align}\tag{1} f(x + h) = f(x) + hf'(x) + \frac{1}...
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Oscillations when solving parabolic heat equation with FTCS

I'm wondering if someone could help me out, or point me in a direction of how I can understand the following oscillations that occur when I solve the Porous Medium Equation $$u_t = u_{xx}^{m+1}$$ ...
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Numerical solution to N-dimensional diffusion on simplex?

Assume I have a system of at least (but generally only) $N+1$ points in an $N$-dimensional space ($N > 3$ is possible). At each of these points $x_i, i=1,...,N+1$ I know an initial potential/...
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125 views

Non-linear Boundary Value Problem. How to compute the Jacobian?

Consider a Boundary Value Problem: $$ \delta u''+u(u'-1) =0 \Leftrightarrow u''=\frac{-u(u'-1)}{\delta}=:f(t,u',u), \\ u(0)=a, u(1)=b $$ $\delta,a,b$ are known parameters. I want to implement Newton'...
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How to make a less diffusive code to solve 2D advection equation?

I would like to solve the following differential equation numerically in 2D, $$\frac{\partial z^-}{\partial t}+(\vec{B}\cdot\vec{\nabla})z^-=0,$$ see Wikipedia if you are curious about what the ...
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1answer
81 views

Actual global error vs theoretical global error: How to combine theory with practice

I have implemented an Adams Bashforth 4 method to solve an Initial Value Problem for an ODE and I am testing it against the test equation: $y'=\lambda y$ with $y(0)=1$ with the exact solution: $y(t)=...
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Boundary Conditions involving exponential functions of nodal unknowns

I am fairly new to Computational Engineering and I have mainly been exposed to using the Finite Difference Method to produce Linear Systems and solve them using Iterative Methods. I am trying to ...
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116 views

Discretization Neumann boundary condition

I'm currently working with the following Poisson equation with mixed boundary conditions, including a Neumann boundary condition. $$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<...
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Numerical integration of Fokker-Planck equation allowing for negative drift?

The Fokker-Planck equation (a.k.a Kolmogorov forward equation or Smoluchowski equation) describes the evolution of a probability density function and numerical integration of the FPE should conserve ...
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Unusual boundary conditions on Matlab

I'm trying to solve the following PDE by Matlab, $$ u_t-\Delta u = 0, \quad \text{in}\quad \Omega\times (0,T) \tag{1} $$ $$ u_t-\Delta_\Gamma u + \partial_\nu u=0,\quad \text{on}\quad \Gamma\times(0,...
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WENO5 scheme in a staggered grid

I'm trying to use the finite-difference WENO scheme to solve the 2D density conservation law with axial symmetry (coordinates $r,z$): $\frac{\partial \rho}{\partial t}+\nabla \cdot (\rho \vec{v}) = \...
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Can I take advantage of a nearly banded A in AX=b?

I am working on a 1D drift-driffusion problem in a finite-difference (FD) approach. I hade 3 equations per node ($3N$ in total): electron continuity $E_i$, Poisson $P_i$, hole continuity $H_i$. With ...
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Hydrogen-like wavefunction as starting guess for atomic solver?

I've been looking into radial solvers for quantum wave equations (Schroedinger and Dirac). In both cases, the suggestion seems to be to go with the "shooting method", with integration schemes of ...
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3answers
207 views

How does a stiff equation solver work?

I am trying to understand how stiff differential equations are solved. For instance the equation, $$\frac{\partial y}{\partial t} = \alpha\frac{\partial ^2 y}{\partial z^2}$$ can be solved using ...
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Python Finite Difference Schemes for 1D Heat Equation: How to express for loop using numpy expression

Hello all, I've recently been introduced to Python and Numpy, and am still a beginner in applying it for numerical methods. I've been performing simple 1D diffusion computations. I suppose my ...
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1answer
69 views

Finite difference - Explicit / Implicit / Crank Nicolson - Does the implicit method require the least memory?

Examine a dynamic 2D heat equation $\dot{u} = \Delta u$ with zero boundary temperature. A standard finite difference approach is used on a rectangle using a $n\times n$ grid. For the resulting linear ...
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1answer
170 views

Analytical Solution of Transport Equation

I'm looking at the analytical solution of the convection-diffusion equation $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$ with initial ...
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2answers
66 views

TVD for temporal dicretisation

I have come across schemes where TVD (with flux limiters) is used for spatial discretisation along with Runge-kutta for Temporal discretisation. Can TVD be used for Temporal discretisation? If so ...
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How to implement register blocking for 3D finite-difference stencil computations

I am in need to optimize the performance of the 3D solver that integrates a system of equations for seismic wave propagation. Unsurprisingly, the function that implements the finite-difference stencil ...
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1answer
61 views

Numerical solution of non-linear first order partial differential equation (HJB)

I am trying to solve a simple optimal control problem using the Hamilton-Jacobi-Bellman equation, numerically in Python. This is proving to be rather difficult as I end up having to solve the ...
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1answer
48 views

Changing the domain of a 3D Finite Difference code from cube to sphere

I have an explicit FD (Finite Difference) code for diffusion/heat on a PDE in a cuboid domain, and it works fine. I would like to update the discretized equations and change the code so as to solve ...