# Tag Info

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This might seem extreme, but this can be analysed exactly. Take the system $$\dot x_1 = x_2, \qquad \dot x_2=-x_1, \qquad x_1(0) = 1, \qquad x_2(0)=0.$$ Let $X=(x_1,x_2)$ be the state vector, $\delta t$ the time step, and $X^+$ the state vector for the next time step. Then the implicit Euler scheme is $$X^+ = \delta t\left(\begin{array}{cc}0&1\\-1&... 12 Short answer If you only want second order accuracy and no embedded error estimation, chances are that you'll be happy with Strang splitting: half-step of reaction, full step of diffusion, half step of reaction. Long answer Reaction-diffusion, even with linear reaction, is famous for demonstrating splitting error. Indeed, it can be much worse, including "... 9 I assume, that you have conducted a space discretization, so that you are about solving the (vector-valued) ODE$$ \dot u_h(t) = F_h(t,u_h(t)), \text{ on [0,T] }, u_h(0) = \alpha. $$via a numerical scheme \Phi that advances the approximation u_h^n at the current time instance t=t^n to the next value u_h^{n+1} at t=t^{n+1}:=t^n+\tau. Then your ... 8 The standard approach is to use a self-starting time-marching algorithm with sufficiently small timestep (such that the order of accuracy is not spoiled) and compute the 5 non-initial value previous solutions. These are then used to "start" the BDF formula. 8 A simplification - the Crank-Nicolson method uses the average of the forward and backward Euler methods. The backward Euler method is implicit, so Crank-Nicolson, having this as one of its components, is also implicit. More accurately, this method is implicit because u^{n+1}_i depends on F^{n+1}_i, not just F^{n}_i This is means that the state at ... 8 If you just slap together an implicit and an explicit method you will likely have order loss. You can do so with low order methods though, and Crank-Nicholson mixed with some other integrator is an easy way to get a decent second order integrator. Higher order IMEX integrators like the Kennedy and Carpenter Additive Runge-Kutta methods or the SBDF schemes ... 6 I think you're mixing up ideas. The best thing here is to know the foundations, and it then "implicit FEM" will be a trivial idea (which is why there won't be a book specifically about that). Finite Element Methods are a form of spatial discretization where you use some kind of structured or unstructured mesh to re-write the problem as a system of equations ... 6 The two properties are usually called positivity-preserving and monotonicity-preserving (makes it easier to find this question). Looking at http://www.ams.org/journals/mcom/2006-75-254/S0025-5718-05-01794-1/S0025-5718-05-01794-1.pdf and http://homepages.cwi.nl/~willem/DOCART/SIAM_HRS.pdf it seems that implicit Euler is the exception among implicit linear ... 6 The reason is that GMRES can only be used for solving linear equations, i.e. equations of the form Ax=b, where A is some matrix and x,b are vectors. What GMRES does, essentially, is it approximates multiplication by the matrix A^{-1} using a matrix polynomial of A. In this case (I assume) f(y^{n+1},t) is not necessarily linear in the vector y^{... 5 The implicit Euler method is unconditionally stable alright, but what you are doing is not the implicit Euler method. Rather, what you do is compute where the particle would be at the end of the time step using only 2-particle interactions, compute the location update, and then sum these updates up for all particle interactions. But the implicit Euler method ... 5 In general, just because method A provides certain guarantees (such as unconditional stability, energy conservation, being symplectic) does not imply that it is more accurate. In fact, a common observation is that the opposite may be true: For example, if a method is symplectic, then it guarantees that the error is zero with regard to certain quantities (e.g.... 5 Your second one is the correct form. BDF approximates derivative with backward difference. Write$$ M(y) \dot{y} = f(y,t) $$as$$ \dot{y} = M^{-1}(y) f(y,t) =: F(y,t) $$write the BDF for this$$ \sum_{k=0}^s \alpha_k y_{n-k} = h F(y_n,t) = h M^{-1}(y_n) f(y_n,t_n) $$Hence you get$$ M(y_n) \sum_{k=0}^s \alpha_k y_{n-k} = h f(y_n,t_n) $$4 This is exactly the case when the lack of information in the question allows to answer it pretty certainly: it is certainly possible. The error would depend on many factors, including the conditioning of the original problem, particular details of the numerical implementation, and chosen simulation parameters. I do not see any contradiction yet. However, I ... 4 To fix notation, denote the Butcher tableau by$$ \begin{array}{c|c} c & A\\ \hline & b^T \end{array} $$where b and c are vectors of length s (the number of stages) and A is a s \times s matrix. Consider the ODE$$ y' = f(t, y) $$and suppose that y_n is a given approximation to y at time t = t_n. A general Runge-Kutta method (... 3 The Crank-Nicolson method is: \frac{u^{n+1}_{i}-u^{n}_{i}}{dt} = \frac{1}{2}(F^{n+1}_{i}+F^{n}_{i}) This method calculates the next state of the system, i.e. u^{n+1}_{i}, by solving an equation involving the previous states and the next state. In the case of the heat equation for example we would get a linear system and if we are using finite elements ... 3 Define$$ \begin{align} A^{n+1} & = e\left[j^{n+1} - f\, θ_H^{n+1}\sinh\left(\frac{g\,n_A^{n+1}}{T}\right)\right] \\ B^{n+1} & = a\left[bP^{n+1}\,(1 - θ_H^{n+1} )^2 - c~(θ_H^{n+1})^2 - f\,θ_H^{n+1}\,\sinh\left(\frac{g\,n_A^{n+1}}{T}\right)\right] \\ C^{n+1} & = h(i - P^{n+1}) + d\,T\,(P^{n+1}-P_a)\left[bP^{n+1}~(1 - θ_H^{n+1} )^2 - c~(θ_H^{...

2

Since pdepe accepts systems of PDEs through vector-valued capacity, flux, and source terms, one way to accommodate your request would be to set the fluxes for all of the $\rho$ variables equal to zero. The capacity terms for the $\rho$ variables will all be 1, and the source terms for each variable are the non-flux terms on the right-hand side (the $\rho$ ...

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As I said in my answer to your previous question, it's probably better if you don't try to write your own ODE solver, which is what you are doing now. There are a lot of good libraries out there that will solve a system of ordinary differential equations much better than if you roll your own implementation. If you can use them, you should; since your ...

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It looks like the model you're trying to solve is: \begin{align} (1/\alpha(w,c))T_{t}(r,t) &= T_{rr}(r,t) + (p/r) \cdot T_{r}(r,t) \\ w_{t}(r,t) &= -(k_{1}(T(r,t)) + k_{2}(T(r,t)) + k_{3}(T(r,t)))w(r,t) \\ g_{t}(r,t) &= k_{1}(T(r,t))w(r,t) \\ a_{t}(r,t) &= k_{2}(T(r,t))w(r,t) \\ c_{t}(r,t) &= k_{3}(T(r,t))w(r,t) \end{align} where: $r =$...

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One of the best ways to test a PDE solver is to use the method of manufactured solutions. Essentially, you modify the PDE (and discretization) by adding a source term that yields an exact solution known in advance. You can then compare your numerical solution against the exact solution for debugging purposes. You should test your numerical solution against a ...

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You cannot merely adjust an explicit RK scheme into an implicit one, implicit routines are much more involved because each intermediate slope can depend upon slopes 'in the future'. This introduces a (possibly nonlinear) system of equations that needs to be solved within each RK step. That is nowhere to be found in the explicit method you posted above. If ...

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Backward Euler and the implicit trapezoidal rule are both unconditionally stable for this problem. If you're seeing instability then you haven't implemented them correctly.

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It really depends on how the matrix will be used. In implicit schemes, you typically solve a system of the form $(I-\gamma A)x=b$, where $\gamma$ is some small number related to a time step. For anything much larger than 1-D and small 2-D problems, you have to think very hard about how to actually store the matrix and solve the problem. Two popular methods ...

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There are many ways to do this. Your "predictor-corrector" approach is one way. A better conceptual framework is to do a time discretization first and then look at what kind of problem you have. For example, if you wanted to do an implicit Euler scheme, you'd have to solve the following problem in each time step: $$\frac{\rho^n-\rho^{n-1}}{\Delta ... 2 Short Answer: There is no general result that would hold for all implicit schemes. The reason is that how your method behaves with respect to numerical diffusion depends on the specific combination of numerical time stepping scheme and spatial finite difference. However, note that if your discretisation is consistent with your PDE and your PDE does not have ... 1 Yes, there is active research on that, especially in the Lyapunov case: it turns out under some conditions M is well approximated by a low-rank or banded matrix. So you can find an implicit representation of the solution M and then use it to compute matrix-vector products, in your case. A good starting point is Valeria Simoncini's research; for instance,... 1 You can solve your problem in two ways: explicitly or implicitly. If you want to solve your system explicitly, then the solution is quite simple: the only term you would be solving for is the u_{i+1} term in the time derivative. All the other instances of u are known quantities (u_{i}), which means they can be placed on the RHS - because they are ... 1 In the computer graphics community there exist many approaches that approximate the advection operator of the Navier-Stokes or Euler equations with unconditional stability by particle tracing that are somehow implicit in time. The basic idea is to use a negative time steps and therefore trace particles backwards in time. At the grid points, where velocity ... 1 In your expression for S, is x = k_x \Delta x and y = k_y \Delta y for some "wave number" vector \mathbf{k} = [k_x \quad k_y]^T with \Delta x  and \Delta y being grid spacings in x and y directions? I assumed that x = \pi and y = \pi and obtained the following for S:$$ -{\frac {2\,cdp+{c}^{2}+{d}^{2}-c-d-3}{{c}^{2}+2\,cd+{d}^{2}+5\,c+...

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The implicit/explicit designation refers to the finite difference time stepping scheme used to compute dynamic models. Nonlinear Finite Element Methods by Peter Wriggers talks about the two different methods, and virtually any textbook on numerical methods for partial differential equations will discuss the difference as well. There are a wide range of ...

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